If tan⁡ x=−15/8, and 3π/2<x<2π, find cos⁡(x−π/4).

Question

is there a basic formula for these kinds of problems i have alot of varying ones i need to do but i dont know the formula(s)

Answer 1

Some basic knowledge is

1) location of 3π / 2 < θ < 2π, that is QIV (I use "x" for the abscissa value)

2) tan θ = -15 / 8, then x = 8 and y = -15 )draw a diagram!)

3) Pythagoras determines h = 17

4) sin θ = y / h, cos θ = x / h

Answer 2

you are in the 4th quadrant

tan is opposite/adjacent

cos is positive and sin is negative

∏/4=45°

draw a right triangle (15 and 8 are the legs)

use the Pythagorean Theorem to find the hypotenuse

(-15)^2+8^2=c^2

225+64=289

c^2=289

take the square root of both sides

c=17, so the hypotenuse is 17

sin=opposite/hypotenuse and cos=adjacent/hypotenuse

sinx=-15/17 and cosx=8/17

use the following trig identity...

cos(a-b)=cosacosb+sinasinb

cos(x-∏/4)=cosxcos45°+sinxsin45°

now substitute the values for sin and cos

(8/17)(√2/2)+(-15/17)(√2/2)= (cos45° and sin45° both equal √2/2)

(8√2/34)-(15√2/34)=

-7√2/34 answer