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Fundamental identities

We can derive some useful identities    from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

tan t = sin t cos t
sec t = 1 cos t
csc t = 1 sin t
cot t = 1 tan t = cos t sin t

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 To divide the functions, we multiply by the reciprocal . = 1 cos t cos t sin t Divide out the cosines . = 1 sin t Simplify and use the identity . = csc t

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 alternative 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.

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 in the chapter opening, 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.

Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
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ismail Reply
What do you need help with?
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ismail
Rectangle coordinate
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Melissa
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mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
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Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Robert
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Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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