| 单词 | cosine |
| 例句 | The program took control of Parzival’s movements, synching them up with the music, and all four of my limbs were transformed into undulating cosine waves. Ready Player One: A Novel 2011-08-16T00:00:00Z I sometimes dreamed about sine, cosine and tangent, about mysterious angles and concussed computations, but for all this I made no real progress. Educated 2018-02-20T00:00:00Z She smoked and talked and ate for an hour without stopping, and I scribbled in my notebook as the muddy waters of tangents and cosines began to clarify. Looking for Alaska 2005-03-03T00:00:00Z The misery began when I moved beyond the Pythagorean theorem to sine, cosine and tangent. Educated 2018-02-20T00:00:00Z The games and displays are so attractive and such fun that the kids bounced from one to the next, not always fully taking in the sines and cosines, the vectors and hypotenuses. Hands-On Math Lessons (No Homework Required) 2013-08-29T21:33:42Z Sure, there is some daunting talk of string theory and convergent series and the cosine of half pi Z, not to mention the varieties of infinity. | 'A Disappearing Number': Simon McBurney?s Play at Lincoln Center Festival 2010-07-16T22:29:00Z You can also use the cosine similarity here to compare content you have already seen with other products. How Recommendation Algorithms Work—And Why They May Miss the Mark 2023-08-17T04:00:00Z These functions, such as the sine and cosine, are defined using right triangles. 2 High School Students Prove Pythagorean Theorem. Here's What That Means 2023-04-10T04:00:00Z Simple waves can be modeled mathematically using sine or cosine functions involving the wavelength, amplitude, and frequency of the wave. College Physics for AP Courses 2015-08-12T00:00:00Z The cosine of 90° is 0; the sine of 90° is 1. Precalculus 2021-12-01T00:00:00Z When we take derivatives of polynomials, exponential functions, sines, and cosines, we get polynomials, exponential functions, sines, and cosines. Calculus, Volume 3 2016-03-30T00:00:00Z Figure 2.84 A hyperbolic cosine function forms the shape of a catenary. Calculus, Vol 2 2016-03-30T00:00:00Z The cosine of this angle is the length of the horizontal side divided by the hypotenuse. 2 High School Students Prove Pythagorean Theorem. Here's What That Means 2023-04-10T04:00:00Z A simple wave can be described by an equation involving one sine or cosine function involving the wavelength, amplitude, and frequency of the wave. College Physics for AP Courses 2015-08-12T00:00:00Z Next, we will find the cosine and sine of the reference angle: Precalculus 2021-12-01T00:00:00Z Underdamped systems do oscillate because of the sine and cosine terms in the solution. Calculus, Volume 3 2016-03-30T00:00:00Z If cosine appears in the denominator, then the conic is horizontal. Calculus, Vol 2 2016-03-30T00:00:00Z Table 1.9 shows the values of sine and cosine at the major angles in the first quadrant. Calculus, Volume 1 2016-03-30T00:00:00Z A simple wave can be described by an equation involving one sine or cosine function involving the wavelength, amplitude, and frequency of the wave. College Physics for AP Courses 2015-08-12T00:00:00Z We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle: Precalculus 2021-12-01T00:00:00Z Find the direction cosines for the vector u. Calculus, Volume 3 2016-03-30T00:00:00Z Then the coefficient of the sine or cosine in the denominator is the eccentricity. Calculus, Vol 2 2016-03-30T00:00:00Z From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. Calculus, Volume 1 2016-03-30T00:00:00Z The first cosine term in the expression effectively causes the amplitude to go up and down. College Physics for AP Courses 2015-08-12T00:00:00Z The value of the sine or cosine function of is its value at radians. Precalculus 2021-12-01T00:00:00Z This strategy works because cosine is positive between 0 and π . Calculus, Volume 3 2016-03-30T00:00:00Z Solving either equation for t directly is not advisable because sine and cosine are not one-to-one functions. Calculus, Vol 2 2016-03-30T00:00:00Z The sine, cosine, secant, and cosecant functions have a period of 2π. Calculus, Volume 1 2016-03-30T00:00:00Z We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. Algebra and Trigonometry 2015-02-13T00:00:00Z We can also calculate sines and cosines of the special angles using the Pythagorean Identity and our knowledge of triangles. Precalculus 2021-12-01T00:00:00Z This property is a result of the fact that we can express the dot product in terms of the cosine of the angle formed by two vectors. Calculus, Volume 3 2016-03-30T00:00:00Z These formulas may be derived from the sum-of-angle formulas for sine and cosine. Calculus, Vol 2 2016-03-30T00:00:00Z The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Calculus, Volume 1 2016-03-30T00:00:00Z The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number. Algebra and Trigonometry 2015-02-13T00:00:00Z This tells us that 150° has the same sine and cosine values as 30°, except for the sign. Precalculus 2021-12-01T00:00:00Z Recall from trigonometry that the law of cosines describes the relationship among the side lengths of the triangle and the angle θ. Calculus, Volume 3 2016-03-30T00:00:00Z The first term can be rewritten as a single cosine function. Calculus, Vol 2 2016-03-30T00:00:00Z Simple harmonic motion can be described by using either sine or cosine functions. Calculus, Volume 1 2016-03-30T00:00:00Z What are the ranges of the sine and cosine functions? Algebra and Trigonometry 2015-02-13T00:00:00Z Since 150° is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. Precalculus 2021-12-01T00:00:00Z Using this equation, we can find the cosine of the angle between two nonzero vectors. Calculus, Volume 3 2016-03-30T00:00:00Z The mnemonic SO-CAH-TOA is commonly used to teach trigonometric functions sine, cosine and tangent. California teacher placed on leave for Native American-themed math lesson 2021-10-22T04:00:00Z We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Calculus, Volume 1 2016-03-30T00:00:00Z As shown in Figure 16, angle has the same sine value as angle the cosine values are opposites. Algebra and Trigonometry 2015-02-13T00:00:00Z Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. Precalculus 2021-12-01T00:00:00Z The inverse cosine is unique over this range, so we are then able to determine the measure of the angle θ. Calculus, Volume 3 2016-03-30T00:00:00Z He proposed that all periodic functions — all periodic phenomena — could be understood as sums of sine and cosine waves. The Godmother of the Digital Image 2021-09-14T04:00:00Z Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion. Calculus, Volume 1 2016-03-30T00:00:00Z Angle has the same cosine value as angle the sine values are opposites. Algebra and Trigonometry 2015-02-13T00:00:00Z This is the standard cosine function shifted up three units. Precalculus 2021-12-01T00:00:00Z Direction angles are often calculated by using the dot product and the cosines of the angles, called the direction cosines. Calculus, Volume 3 2016-03-30T00:00:00Z The functions sine, cosine, tangent, exsecant, and so on all refer to ratios of side lengths of right triangles. A Few of My Favorite Spaces: The 1-Dimensional Sphere 2017-10-07T04:00:00Z Figure 3.26 These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions. Calculus, Volume 1 2016-03-30T00:00:00Z To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Algebra and Trigonometry 2015-02-13T00:00:00Z We will use the cosine function, since the spring starts out at its maximum displacement. Precalculus 2021-12-01T00:00:00Z Therefore, we define both these angles and their cosines. Calculus, Volume 3 2016-03-30T00:00:00Z I could see the relationship between sine and cosine just by visualizing an angle moving along the circle. A Few of My Favorite Spaces: The 1-Dimensional Sphere 2017-10-07T04:00:00Z Figure 6.84 A hyperbolic cosine function forms the shape of a catenary. Calculus, Volume 1 2016-03-30T00:00:00Z The absolute values of the cosine and sine of an angle are the same as those of the reference angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Our strategy is to find the sine, cosine, and tangent of the angles first. Precalculus 2021-12-01T00:00:00Z This vector is a unit vector, and the components of the unit vector are called directional cosines. Calculus, Volume 3 2016-03-30T00:00:00Z Trigonometry, the study of the lengths and angles of triangles, sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines. This ancient Babylonian tablet may contain the first evidence of trigonometry 2017-08-24T04:00:00Z At age 16, he wrote a program to calculate trigonometric functions like sine and cosine using a “neural network” - the core computing engine of artificial intelligence modeled on the human brain. Why AI visionary Andrew Ng teaches humans to teach computers 2017-08-21T04:00:00Z The cosine will be positive or negative depending on the sign of the x-values in that quadrant. Algebra and Trigonometry 2015-02-13T00:00:00Z Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent. Precalculus 2021-12-01T00:00:00Z The method of undetermined coefficients also works with products of polynomials, exponentials, sines, and cosines. Calculus, Volume 3 2016-03-30T00:00:00Z The familiar sines, cosines, and angles used by Greek astronomers and modern-day high schoolers were completely missing. This ancient Babylonian tablet may contain the first evidence of trigonometry 2017-08-24T04:00:00Z All detected signatures could be assigned to one of the known signatures with a cosine similarity of at least 0.85. The whole-genome landscape of medulloblastoma subtypes : Nature : Nature Research 2017-07-18T04:00:00Z Since is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. Algebra and Trigonometry 2015-02-13T00:00:00Z The cosine function will apply to both models. Precalculus 2021-12-01T00:00:00Z The cosines of these angles are known as the direction cosines. Calculus, Volume 3 2016-03-30T00:00:00Z The function is a combination of sines and cosines with a lot of sixes and fourteens floating around, but somehow it has five-fold rotational symmetry. Impossible Wallpaper and Mystery Curves: Exploring Symmetry in Mathematics and Art 2015-09-14T04:00:00Z Sine and cosine waves are the most common trigonometric functions. Motion sensors to help students visualize math 2014-05-18T04:00:00Z Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant. Algebra and Trigonometry 2015-02-13T00:00:00Z We see that the cosine function is at the maximum displacement when and the sine function is at the equilibrium point when For example, consider the equation from Example 10. Precalculus 2021-12-01T00:00:00Z The law of cosines can be thought of as a function of three variables. Calculus, Volume 3 2016-03-30T00:00:00Z For several hours, they figure out tangents and cosines, tapping away at their calculators to find the distance of a line, the degree of an angle, or the circumference of a cylinder. Welders, America Needs You 2014-03-21T01:21:01Z The cosine value threshold was set to 0.5, whereas a cosine value of one indicates identical spectra and a cosine value of 0 displays no spectral correlation. An environmental bacterial taxon with a large and distinct metabolic repertoire 2014-02-05T18:21:56.312Z Next, we will find the cosine and sine of the reference angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Check this by setting in the cosine equation: Precalculus 2021-12-01T00:00:00Z Since cosine is negative and sine is positive, the angle must be in the second quadrant. Calculus, Volume 3 2016-03-30T00:00:00Z The familiar sine, cosine, and tangent are in blue, red, and, well, tan, respectively. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z The word has been sometimes, though rarely, used to denote the complement of a given logarithm; also the logarithmic cosine corresponding to a given logarithmic sine. Webster's Unabridged Dictionary 2012-03-24T02:00:23.513Z Because our original angle is in the third quadrant, where both and are negative, both cosine and sine are negative. Algebra and Trigonometry 2015-02-13T00:00:00Z The difference in its magnetic state, at the two periods, is shown by the squares of the times of vibration multiplied by the cosine of the dip observed at the respective periods. Narrative of the surveying voyages of His Majesty's ships Adventure and Beagle, between the years 1826 and 1836 Volume I. - Proceedings of the First Expedition, 1826-1830 2012-02-24T03:00:32.793Z The second, or harmonic method, leads to a series of terms involving sines and cosines, the coefficients of which have to be determined. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 8 "Haller, Albrecht" to "Harmonium" 2012-01-02T03:00:22.443Z They’re all just simple combinations of dear old sine and cosine. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z And the next thing I knew I was in a trigonometry class and they’re talking about sines and cosines without any mention that it had something to do with triangles. Marie Curie, Theater, and Science Communication: An Interview with Alan Alda 2011-12-01T16:45:00.263Z Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle. Algebra and Trigonometry 2015-02-13T00:00:00Z If the coordinates are Cartesian and l, m, n are direction cosines, the quantity on the left is the mutual moment of the two lines. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Elliptic Functions are generalizations of the circular functions sine, cosine, &c. The New Gresham Encyclopedia Volume 4, Part 2: Ebert to Estremadura 2011-04-14T02:00:56.200Z If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z Also let e denote the extension in the direction of a line the direction cosines of which are l, m, n. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 2 "Ehud" to "Electroscope" 2011-01-29T03:00:23.777Z For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by lies. Algebra and Trigonometry 2015-02-13T00:00:00Z By referring to a table of cosines, we find that 0.99984 is the cosine of 1 degree, approximately; hence, the taper attachment slide and the work should be set to this angle. Turning and Boring A specialized treatise for machinists, students in the industrial and engineering schools, and apprentices, on turning and boring methods, etc. Let F, G, H be the components per unit area of the force on the plane whose direction cosines, drawn outwards, are l, m, n. The New Gresham Encyclopedia Volume 4, Part 2: Ebert to Estremadura 2011-04-14T02:00:56.200Z If you have a computation involving the square of sine or cosine, you can use a haversine or havercosine table and not have to square or take square roots. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z For mercury in a glass tube the angle of contact is 128� 52′, the cosine of which is negative. Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 3 "Capefigue" to "Carneades" For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle? Algebra and Trigonometry 2015-02-13T00:00:00Z His father stood over him in wonder while he filled a sheet of paper with sines, cosines, secants, and such things. Harper's Round Table, June 4, 1895 "But that ain't right," said the other; "I say it is the cosine." Books and Authors Curious Facts and Characteristic Sketches A diagram showing the sine, cosine, and versine of an angle. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z If six films of the same liquid meet in a point the corresponding tetrahedron is a regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is −1⁄3. Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 3 "Capefigue" to "Carneades" The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. Algebra and Trigonometry 2015-02-13T00:00:00Z Coming out of the class-room one day, her heart singing of sines and cosines and tangents, Peggy found the Snowy and the Fluffy waiting for her at the door, with radiant faces. Peggy Radius of rotation here in Washington would be cosine latitude times equatorial radius, approximately—call it thirty-two hundred miles. Skylark Three For instance, the cosine of an angle is also the sine of the complementary angle. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z In order to make any solution at all possible you have assumed cosine squared theta negligible. Spacehounds of IPC When the sine or cosine is known, we can use the Pythagorean Identity to find the other. Algebra and Trigonometry 2015-02-13T00:00:00Z Had the earth no obliquity, the effect would be as the squares of the cosines of the latitude; but the ratio is diminished by the inclination of the axis. Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence Secant equals one over cosine—um-m-m-m—one point oh three five. Skylark Three My guess is that vercosine was a later term, an analogy of the square of sine definition of versine using cosine instead. 10 Secret Trig Functions Your Math Teachers Never Taught You 2013-09-16T13:18:09.820Z Mathematically, it is of course vanishingly small compared to the first power of the cosine, but fields of that type must be exact, and your neglect of the square is indefensible. Spacehounds of IPC State the domain of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z In other latitudes it is diminished in the ratio of the squares of the cosines of the latitude; it therefore becomes 1 ⁄ 434 in that latitude the square of whose sine is ⅓. Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence If he did, he'd probably make another pun about sines and cosines. The Radio Boys in the Thousand Islands So it goes on until our parents begin to look up to us because we can chatter of cosines or sketch the foreign policy of Louis XIV. The Human Machine Let them bring their compasses, or some other instrument for measuring the angles, sines and cosines of human conditions. A Walk from London to John O'Groat's Use phase shifts of sine and cosine curves. Algebra and Trigonometry 2015-02-13T00:00:00Z The difference between the cosine of 13° and radius applied to the circumference, is about 600 miles, which would give 25 miles per hour to the eastward, in lat. Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence Tables of generalized sine—and cosine—integral functions. U.S. Copyright Renewals, 1977 January - June C E is the sine, O E is the cosine, and E A is the versed sine of the angle A O C. Respecting these three lines there are many things to be observed. Scientific American Supplement, No. 531, March 6, 1886 Now, gentlemen cosines, will you cease to throw parabolas and hyperbolas at each other's heads? From the Earth to the Moon; and, Round the Moon In this section, we will interpret and create graphs of sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Multiply this by the cosine of the angle and divide by the sine, and we shall get the value of the arc QY for the equator. Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence Tables of generalized sine—and cosine—integral functions. U.S. Copyright Renewals, 1977 January - June If, finally, I make O C to coincide with O A, the angle is obliterated, the sine and the versed sine have both disappeared, and the cosine has become the radius. Scientific American Supplement, No. 531, March 6, 1886 Nevertheless, I bear no grudge against the sine and the cosine, which I continue to hold in high esteem. The Life of the fly; with which are interspersed some chapters of autobiography TRY IT #1 Given write a relation involving the inverse cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z McTurk may have some conception of the meaning of it; but Beetle is as the brutes that perish about sines and cosines. Stalky & Co. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The sine and the cosine I shall have occasion to employ in the latter part of my lecture. Scientific American Supplement, No. 531, March 6, 1886 I confess that I fumed against the regulation which forced Latin and Greek upon me before allowing me to open up relations with the sine and cosine. The Life of the fly; with which are interspersed some chapters of autobiography Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. Algebra and Trigonometry 2015-02-13T00:00:00Z A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. Algebra and Trigonometry 2015-02-13T00:00:00Z The graph could represent either a sine or a cosine function that is shifted and/or reflected. Algebra and Trigonometry 2015-02-13T00:00:00Z When the graph has an extreme point, Since the cosine function has an extreme point for let us write our equation in terms of a cosine function. Algebra and Trigonometry 2015-02-13T00:00:00Z The other even-odd identities follow from the even and odd nature of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Because we know that the inverse sine must give an angle on the interval we can deduce that the cosine of that angle must be positive. Algebra and Trigonometry 2015-02-13T00:00:00Z The cotangent identity, also follows from the sine and cosine identities. Algebra and Trigonometry 2015-02-13T00:00:00Z So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. Algebra and Trigonometry 2015-02-13T00:00:00Z Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. Algebra and Trigonometry 2015-02-13T00:00:00Z In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent. Algebra and Trigonometry 2015-02-13T00:00:00Z The graph of a sinusoidal function has the same general shape as a sine or cosine function. Algebra and Trigonometry 2015-02-13T00:00:00Z What would you estimate the cosine of degrees to be? Algebra and Trigonometry 2015-02-13T00:00:00Z If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. Algebra and Trigonometry 2015-02-13T00:00:00Z The Pythagorean Identity is also useful for determining the sines and cosines of special angles. Algebra and Trigonometry 2015-02-13T00:00:00Z Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. Algebra and Trigonometry 2015-02-13T00:00:00Z The identity is found by rewriting the left side of the equation in terms of sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z The domain of the sine and cosine functions is all real numbers. Algebra and Trigonometry 2015-02-13T00:00:00Z The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the original angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Reference angles can be used to find the sine and cosine of the original angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Table 2 lists some of the values for the cosine function on a unit circle. Algebra and Trigonometry 2015-02-13T00:00:00Z Similarly, can be obtained by rewriting the left side of this identity in terms of sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z Since is negative, the graph of the cosine function has been reflected about the x-axis. Algebra and Trigonometry 2015-02-13T00:00:00Z As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function. Algebra and Trigonometry 2015-02-13T00:00:00Z Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. Algebra and Trigonometry 2015-02-13T00:00:00Z A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Algebra and Trigonometry 2015-02-13T00:00:00Z In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve. Algebra and Trigonometry 2015-02-13T00:00:00Z Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30. Algebra and Trigonometry 2015-02-13T00:00:00Z Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Algebra and Trigonometry 2015-02-13T00:00:00Z Let’s investigate the cosine identity first and then the sine identity. Algebra and Trigonometry 2015-02-13T00:00:00Z Note that, because cosine is an even function, secant is also an even function. Algebra and Trigonometry 2015-02-13T00:00:00Z Find the exact value of expressions involving the inverse sine, cosine, and tangent functions. Algebra and Trigonometry 2015-02-13T00:00:00Z In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. Algebra and Trigonometry 2015-02-13T00:00:00Z Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. Algebra and Trigonometry 2015-02-13T00:00:00Z Then, we divide by to isolate the product of cosines: Algebra and Trigonometry 2015-02-13T00:00:00Z Then, we divide by 2 to isolate the product of cosine and sine: Algebra and Trigonometry 2015-02-13T00:00:00Z In this case, we will first subtract the two cosine formulas: Algebra and Trigonometry 2015-02-13T00:00:00Z Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas. Algebra and Trigonometry 2015-02-13T00:00:00Z The sum-to-product formulas allow us to express sums of sine or cosine as products. Algebra and Trigonometry 2015-02-13T00:00:00Z We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. Algebra and Trigonometry 2015-02-13T00:00:00Z Give your answer in terms of sines and cosines. Algebra and Trigonometry 2015-02-13T00:00:00Z The period of both the sine function and the cosine function is In other words, every units, the y-values repeat. Algebra and Trigonometry 2015-02-13T00:00:00Z We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions? Algebra and Trigonometry 2015-02-13T00:00:00Z We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. Algebra and Trigonometry 2015-02-13T00:00:00Z The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles. Algebra and Trigonometry 2015-02-13T00:00:00Z The identities can be verified using other formulas or by converting the expressions to sines and cosines. Algebra and Trigonometry 2015-02-13T00:00:00Z We have already defined the sine and cosine functions of an angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Though sine and cosine are the trigonometric functions most often used, there are four others. Algebra and Trigonometry 2015-02-13T00:00:00Z A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Algebra and Trigonometry 2015-02-13T00:00:00Z As with the sine and cosine, we can use the coordinates to find the other functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The secant function is the reciprocal of the cosine function. Algebra and Trigonometry 2015-02-13T00:00:00Z Finally, in quadrant IV, “Class,” only cosine and its reciprocal function, secant, are positive. Algebra and Trigonometry 2015-02-13T00:00:00Z For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z Figure 9 The sine of equals the cosine of and vice versa. Algebra and Trigonometry 2015-02-13T00:00:00Z For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or will result in the same outputs for these functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative. Algebra and Trigonometry 2015-02-13T00:00:00Z We can also calculate sines and cosines of the special angles using the Pythagorean Identity. Algebra and Trigonometry 2015-02-13T00:00:00Z What are the domains of the sine and cosine functions? Algebra and Trigonometry 2015-02-13T00:00:00Z The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. Algebra and Trigonometry 2015-02-13T00:00:00Z Therefore, its cosine value will be the opposite of the first angle’s cosine value. Algebra and Trigonometry 2015-02-13T00:00:00Z Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. Algebra and Trigonometry 2015-02-13T00:00:00Z Its reference angle is The cosine and sine of are both In the third quadrant, both and are negative, so: Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. Algebra and Trigonometry 2015-02-13T00:00:00Z The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z State the range of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 27. Algebra and Trigonometry 2015-02-13T00:00:00Z Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28. Algebra and Trigonometry 2015-02-13T00:00:00Z Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32. Algebra and Trigonometry 2015-02-13T00:00:00Z State the domain of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value. Algebra and Trigonometry 2015-02-13T00:00:00Z We can graph by observing the graph of the cosine function because these two functions are reciprocals of one another. Algebra and Trigonometry 2015-02-13T00:00:00Z Where the graph of the cosine function decreases, the graph of the secant function increases. Algebra and Trigonometry 2015-02-13T00:00:00Z Where the graph of the cosine function increases, the graph of the secant function decreases. Algebra and Trigonometry 2015-02-13T00:00:00Z State the range of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z When the cosine function is zero, the secant is undefined. Algebra and Trigonometry 2015-02-13T00:00:00Z Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function. Algebra and Trigonometry 2015-02-13T00:00:00Z Again, substitute the values of the sine and cosine into the equation, and simplify. Algebra and Trigonometry 2015-02-13T00:00:00Z They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z State the range of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z The last reduction formula is derived by writing tangent in terms of sine and cosine: Algebra and Trigonometry 2015-02-13T00:00:00Z Then we substitute the value of the cosine we found from the triangle in Figure 3 and simplify. Algebra and Trigonometry 2015-02-13T00:00:00Z Again, we substitute the value of the cosine we found from the triangle in Figure 3 and simplify. Algebra and Trigonometry 2015-02-13T00:00:00Z In this chapter, we will investigate graphs of sine, cosine, and other trigonometric functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. Algebra and Trigonometry 2015-02-13T00:00:00Z The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The cosine of is 0; the sine of is 1. Algebra and Trigonometry 2015-02-13T00:00:00Z Because the x- and y-values are the same, the sine and cosine values will also be equal. Algebra and Trigonometry 2015-02-13T00:00:00Z As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4. Algebra and Trigonometry 2015-02-13T00:00:00Z Using the Pythagorean Identity, we can find the cosine value. Algebra and Trigonometry 2015-02-13T00:00:00Z Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period. Algebra and Trigonometry 2015-02-13T00:00:00Z To sum up, only two of the trigonometric functions, cosine and secant, are even. Algebra and Trigonometry 2015-02-13T00:00:00Z First, we will prove the difference formula for cosines. Algebra and Trigonometry 2015-02-13T00:00:00Z Figure 5 shows several periods of the sine and cosine functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Thus, we have the difference formula for cosine. Algebra and Trigonometry 2015-02-13T00:00:00Z We can use similar methods to derive the cosine of the sum of two angles. Algebra and Trigonometry 2015-02-13T00:00:00Z Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Algebra and Trigonometry 2015-02-13T00:00:00Z Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. Algebra and Trigonometry 2015-02-13T00:00:00Z Again, we determined that the cosine function is an even function. Algebra and Trigonometry 2015-02-13T00:00:00Z As it is simpler to solve for one trigonometric function at a time, we will choose the double-angle identity involving only cosine: Algebra and Trigonometry 2015-02-13T00:00:00Z Each one of the three laws of cosines begins with the square of an unknown side opposite a known angle. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, suppose that represents the relationship of three sides of a triangle and the cosine of an angle. Algebra and Trigonometry 2015-02-13T00:00:00Z |
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