Use identities to find the exact value for each requested function.

You can use the Pythagorean, Tangent and Reciprocal Identities to find all six trigonometric values for certain angles. Let’s walk through a few problems so that you understand how to do this.

Let's solve the following problems using trigonometric identities.

Given that cosθ=35 and 0<θ<π2, find sinθ.

Use the Pythagorean Identity to find sinθ.

sin2θ+cos2θsin2θ+(35)2sin2θsin2θsinθ=1=1=1−925=1625=±45

Because θ is in the first quadrant, we know that sine will be positive. sinθ=45

Find tanθ from #1 above.

Use the Tangent Identity to find tanθ.

tanθ=sinθcosθ=4535=43

Find the other three trigonometric functions of θ from #1.

To find secant, cosecant, and cotangent use the Reciprocal Identities.

cscθ=1sinθ=145=54secθ=1cosθ=135=53cotθ=1tanθ=143=34

ExamplesExample 1

Earlier, you were asked to find cosθ and tanθ of sinθ=23,π2<θ<π.

First, use the Pythagorean Identity to find cosθ.

sin2θ+cos2θ(23)2+cos2θ=1cos2θcos2θcosθ=1=1−49=59=±5√3

However, because θ is restricted to the second quadrant, the cosine must be negative. Therefore, cosθ=−5√3.

Now use the Tangent Identity to find tanθ.

tanθ=sinθcosθ=23−5√3=−25√=−25√5

Find the values of the other five trigonometric functions.

Example 2

tanθ=−512,π2<θ<π

First, we know that θ is in the second quadrant, making sine positive and cosine negative. For this problem, we will use the Pythagorean Identity 1+tan2θ=sec2θ to find secant.

1+(−512)21+25144169144±1312−1312=sec2θ=sec2θ=sec2θ=secθ=secθ

If secθ=−1312, then cosθ=−1213. sinθ=513 because the numerator value of tangent is the sine and it has the same denominator value as cosine. cscθ=135 and cotθ=−125 from the Reciprocal Identities.

Example 3

cscθ=−8,π<θ<3π2

θ is in the third quadrant, so both sine and cosine are negative. The reciprocal of cscθ=−8, will give us sinθ=−18. Now, use the Pythagorean Identity sin2θ+cos2θ=1 to find cosine.

(−18)2+cos2θcos2θcos2θcosθcosθ=1=1−164=6364=±37√8=−37√8

secθ=−837√=−87√21,tanθ=137√=7√21, and cotθ=37√

ReviewIn which quadrants is the sine value positive? Negative?In which quadrants is the cosine value positive? Negative?In which quadrants is the tangent value positive? Negative?

Find the values of the other five trigonometric functions of θ.

sinθ=817,0<θ<π2cosθ=−56,π2<θ<πtanθ=3√4,0<θ<π2secθ=−419,π<θ<3π2sinθ=−1114,3π2<θ<2πcosθ=2√2,0<θ<π2cotθ=5√,π<θ<3π2cscθ=4,π2<θ<πtanθ=−710,3π2<θ<2πAside from using the identities, how else can you find the values of the other five trigonometric functions?Given that cosθ=611 and θ is in the 2nd quadrant, what is sin(−θ)?Given that tanθ=−58 and θ is in the 4th quadrant, what is sec(−θ)?