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

You are given the following information about θ

sinθ=23,π2<θ<π

What are cosθ and tanθ?

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.

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

Use the Tangent Identity to find tanθ.

tanθ=sinθcosθ=4535=43

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

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

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.

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.

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√

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

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