Approximate the following sum as a single sine function. −5 sin(x) + 12 cos(x)

Question

please show each step and how it works

Answer 1

Multiply and divide by √[(-5)²+(12)²] = 13 to get:

13[(-5/13)sinx + (12/13)cosx] = 13sin(x+θ), where cosθ = -5/13 and sinθ = 12/13 (find θ)

Answer 2

-

It's not clear what the problem is, but as stated:

-5sinx + 12cosx

sin^2x + cos^2x = 1

cos^2x = 1- sin^2x

-5sinx + 12cosx

= -5sinx + 12(+ sqr(1-sin^2x)

or notice there's a 5-12-13 right triangle in the 4th quadrant

13^2 = -5^2 + 12^2

169 = 25 + 144

sinx = cosx

x = 45 degrees or pi/4

sin45 = cos45 = sqr2/2

-5sin45 + 12cos45

= -5(sqr2/2) + 12(sqr2/2)

= 7(sqr2/2) = about 7(.707)

= 4.9497 = about 5