Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x) < 0

Question

found cos2x but cant seem to get the other two right and i dont know what im doing wrong

Answer 1

Greetings! Lets solve this shall we ?

So, we must find all three Double-Angle Formulas cos(2x), sin(2x) and tan(2x). Let us begin with finding sin(2x). We have the given fact that cscx = 8. Then, this means (1/sinx) = 8. Lets take the reciprocal of both sides to obtain sinx = 1/8! Now, we can find cosx using the Pythagorean Theorem such that

x² + 1² = 8² .......Subtract 1 from both sides to get

x² = 64-1 = 63......Then square root both sides to get

x = √63 = √(9*7) = 3√7

Then, cosx = (3√7 / 8). Let sin(2x) = 2sinxcosx such that

sin2x = 2(1/8)(3√7/8) = (6√7 / 64) = (3√7 / 32)

Now, lets find cos2x such that cos2x = cos²x - sin²x. Then we plug in sinx and cosx to get

cos2x = (3√7 / 8)² - (1/8)² = (9*7 / 64) - (1 / 64) = (63/64) - (1/64) = (62/64) = 31/32.

Finally, we obtain tan2x such that tan2x = (2tanx) / (1-tan²x), where tanx = (sinx/cosx) = (1/8) / (3√7/8) = (1/8)*(8/3√7) = (1/3√7) = (√7/21) (after rationalizing the denominator, multiplying numerator and denominator by √7). Then,

tan2x = 2(√7/21) / (1-(√7/21)²) = (2√7/21) / (1-(7/441)) = (2√7/21) / (434/441) = (2√7/21)*(441/434) = (21√7/217) = 3√7/31

I hope this helped!