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# Trigonometry Help ASAP ?

Given sinx = 2/3 and cos < 0

a) Use double identities to find the exact values of sin2x and cos2x

I got cos2x =1/9 and sin2x =4\sqrt(5)/9

I'm just confused on how to do b & c.

b) Use a Pythagorean Identity to verify the answers you found for sin2x and cos2x.

c) Use a Ratio Identity to find tan2x. Do not use the double angel identity for tangent.

### 2 Answers

- Wayne DeguManLv 72 months ago
sinx = 2/3 and cosx < 0...hence, x is in Qii

a) sin2x = 2sinxcosx

Now, sin²x + cos²x = 1

so, cos²x = 1 - sin²x

i.e. cos²x = 1 - (2/3)² => 5/9

Hence, cosx = ±√5/3

Then, cosx = -√5/3...as x is in Qii

so, sin2x = 2(2/3)(-√5/3) => -4√5/9

Note: your result of 4\√5/9...is not correct

Now, cos2x = 1 - 2sin²x

so, cos2x = 1 - 2(2/3)² => 1/9

Note: cos2x > 0 becuase if x is in Qii, then 2x is in Qiv where cos2x > 0

b) We already know that sin²x + cos²x = 1

so, sin²2x + cos²2x = 1

i.e. (-4√5/9)² + (1/9)² => 80/81 + 1/81 = 1

c) tan2x = sin2x/cos2x => (-4√5/9)/(1/9)

i.e. tan2x = -4√5

Note: if x is in Qii, then 2x is in Qiv where tan2x < 0

:)>

- ted sLv 72 months ago
so the reference triangle for x is { √5 , 2 , 3 } ===> cos 2x = 5/9 - 4 / 9 = 1 / 9 ====> reference triangle for 2x is { 1 , √80 , 9 } ===> sin 2x = √80 / 9 and tan 2x = √80....b) & c) contained in this work