Trig identities

Question

Suppose that cos𝑥=8/17 and 𝑥 are in the fourth quadrant and that sin𝑦=24/25 and 𝑦 is in the second quadrant. 

Which trig identity would you use for sin(x + y)?

Which trig identity would you use for tan (x - y)?

Answer 1

sinx = - √1 - 64/289 = - 15/17; cosy = - √1 - 576/625 = - 7/25; tanx = sinx/cosx = - 15/8; tany = - 24/7

sin(x + y) = sinxcosy + sinycosx = - 15/17·(- 7/25) + 8/17·24/25 = (105 + 192)/(17·25) = 297/425

tan(x - y) = (tanx - tany)/(1 + tanx·tany) = (- 15/8 - 24/7)-/(1 + 15/8·24/7) = -- 297/416

Answer 2

happy holiday

trigonometric sum of angles formulas like sin(x+y)=sin(x)*cos(y)+sin(y)*cos(x)

and tan(x-y)=(tan(x)-tan(y))/(1+tan(x)*tan(y))

are identities in that for any x and y that are real numbers the equation are always true

but to make clear the equations are super important for angle sums we also call them formulas.

use them for your problem and when you see trig sum formulas, those are two of them.

Important note sin(x)=-√(1-(8/17)²)=[-√(17²-8²)]/17 note also sin(x) in fourth quadrant is negative (sin(A)=+/-√(1-cos²(A)) another definition/identity).

similarly cos(y)=[-√(25²-24²)]/25 where cos(y) in second quadrant is negative.

tan(A)=sin(A)/cos(A) is a definition and therefore an identity, so tan(x)=(-√(17²-8²)(17*8/17)=(-√(17²-8²))/8

and tan(y)=(24/25)/(-√(25²-24²)/25=24/(-√(25²-24²),

be careful with all the algebra