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Using identities to evaluate trigonometric functions

  1. Given sin ( 45° ) = 2 2 , cos ( 45° ) = 2 2 , evaluate tan ( 45° ) .
  2. Given sin ( 5 π 6 ) = 1 2 , cos ( 5 π 6 ) = 3 2 , evaluate sec ( 5 π 6 ) .

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.


  1. tan ( 45° ) = sin ( 45° ) cos ( 45° ) = 2 2 2 2 = 1

  2. sec ( 5 π 6 ) = 1 cos ( 5 π 6 ) = 1 3 2 = −2 3 1 = −2 3 = 2 3 3
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Evaluate csc ( 7 π 6 ) .

2

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Using identities to simplify trigonometric expressions

Simplify sec t tan t .

We can simplify this by rewriting both functions in terms of sine and cosine.

sec  t tan  t = 1 cos  t sin  t cos  t = 1 cos  t   cos  t sin  t Multiply by the reciprocal . = 1 sin  t = csc  t Simplify and use the identity .

By showing that sec t tan t can be simplified to csc t , we have, in fact, established a new identity.

sec t tan t = csc t
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Simplify ( tan t ) ( cos t ) .

sin t

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Alternate forms of the pythagorean identity

We can use these fundamental identities to derive alternate forms of the Pythagorean Identity, cos 2 t + sin 2 t = 1. One form is obtained by dividing both sides by cos 2 t .

cos 2 t cos 2 t + sin 2 t cos 2 t = 1 cos 2 t 1 + tan 2 t = sec 2 t

The other form is obtained by dividing both sides by sin 2 t .

cos 2 t sin 2 t + sin 2 t sin 2 t = 1 sin 2 t cot 2 t + 1 = csc 2 t

Alternate forms of the pythagorean identity

1 + tan 2 t = sec 2 t
cot 2 t + 1 = csc 2 t

Using identities to relate trigonometric functions

If cos ( t ) = 12 13 and t is in quadrant IV, as shown in [link] , find the values of the other five trigonometric functions.

This is an image of graph of circle with angle of t inscribed. Point of (12/13, y) is at intersection of terminal side of angle and edge of circle.

We can find the sine using the Pythagorean Identity, cos 2 t + sin 2 t = 1 , and the remaining functions by relating them to sine and cosine.

( 12 13 ) 2 + sin 2 t = 1 sin 2 t = 1 ( 12 13 ) 2 sin 2 t = 1 144 169 sin 2 t = 25 169 sin  t = ± 25 169 sin  t = ± 25 169 sin  t = ± 5 13

The sign of the sine depends on the y -values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y -values are negative, its sine is negative, 5 13 .

The remaining functions can be calculated using identities relating them to sine and cosine.

tan  t = sin  t cos  t = 5 13 12 13 = 5 12 sec  t = 1 cos  t = 1 12 13 = 13 12 csc  t = 1 sin  t = 1 5 13 = 13 5 cot  t = 1 tan  t = 1 5 12 = 12 5
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If sec ( t ) = 17 8 and 0 < t < π , find the values of the other five functions.

cos t = 8 17 ,   sin t = 15 17 ,   tan t = 15 8 csc t = 17 15 ,   cot t = 8 15

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As we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2 π , will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Other functions can also be periodic. For example, the lengths of months repeat every four years. If x represents the length time, measured in years, and f ( x ) represents the number of days in February, then f ( x + 4 ) = f ( x ) . This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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