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.

normally working animals are domesticated and ruled by a master, a domesticator (tamer), to perform basic and important functions on a farm, for example. the value used in this context is animal labor in exchange for food. it is not so much a contrast to the story told by napoleon and the pigs, since the animals of that particular farm also performed functions on the farm through domestication. the only thing is that in this case, who rules them is a pig, also an animal, but very respected, to whom the other animals lent him submission. that's about a strong relationship of dependency between animal and owner.