SOLUTION: Using the pythagorean identities, if sin^2 is 4/7, then what is tan^2

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Question 858308: Using the pythagorean identities, if sin^2 is 4/7, then what is tan^2
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--

Problem:
Using Pythagorean Identities, is sin%5E2%28x%29=4%2F7, then what is tan%5E2%28x%29?

Solution:
We have three forms of the Pythagorean Identity:
I. sin%5E2%28x%29%2Bcos%5E2%28x%29=1
II. tan%5E2%28x%29%2B1=sec%5E2%28x%29
III. 1%2Bcot%5E2%28x%29=csc%5E2%28x%29

We will make a series of substitutions in order to get tan%5E2%28x%29 in terms of sin%5E2%28x%29.

Rearrange Identity I,
cos%5E2%28x%29=1+-+sin%5E2%28x%29

If a = b, then 1/a=1/b. So
1%2Fcos%5E2%28x%29+=+1%2F%281-sin%5E2%28x%29%29

By definition, 1/cos(x)=sec(x). So,
sec%5E2%28x%29+=+1%2F%281-sin%5E2%28x%29%29

By Identity II, substitute tan%5E2%28x%29%2B1 for sec%5E2%28x%29
tan%5E2%28x%29%2B1+=+1%2F%281-sin%5E2%28x%29%29

Subtract 1 from both sides to isolate tan%5E2%28x%29 on the left side.
tan%5E2%28x%29+=+1%2F%281-sin%5E2%28x%29%29-1

Substitute 4/7 for sin%5E2%28x%29.
tan%5E2%28x%29+=+1%2F%281-4%2F7%29-1

Simplify.
tan%5E2%28x%29+=+7%2F3+-+3%2F3
tan%5E2%28x%29+=+4%2F3

Why can CHECK OUR ANSWER using Identity I and the definition of tangent.
sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1
cos%5E2%28x%29+=+1+-+sin%5E2%28x%29

Substitute 4/7 for sin%5E2%28x%29 and simplify.
cos%5E2%28x%29+=+1+-+4%2F7
cos%5E2%28x%29+=+3%2F7

By definition tan(x) = sin(x) / cos(x). So,
tan%5E2%28x%29=sin%5E2%28x%29%2Fcos%5E2%28x%29

Substitute 4/7 for sin%5E2%28x%29 and 3/7 for cos%5E2%28x%29.
tan%5E2%28x%29=%284%2F7%29%2F%283%2F7%29

Simplify.
tan%5E2%28x%29=%284%2F7%29%2A%287%2F3%29=4%2F3

I hope this helps. Feel free to email me if you have any questions about these steps.

Mrs. Figgy
math.in.the.vortex@gmail.com