solve the equation for a value of x in the first quadrant

Question

6cos(6x)+5=4cos(6x)+6

give an answer in radians and degrees rounded to three decimals.

Answer 1

Given 6 cos (6x) + 5 = 4 cos (6x) + 6, gather variable terms on the left side and simplify

6 cos (6x) - 4 cos (6x) = 6 - 5

2 cos (6x) = 1

Divide both sides by 2

cos (6x) = 1/2

In the first quadrant (0 < θ < pi/2), the only angle with cosine = 1/2 is pi/3 (that is, 60°).

Therefore:

if 6x = pi/3,

then x = pi/18 (or x = 3.333°).

Checking:

6 cos (6x) + 5 = 4 cos (6x) + 6

6 cos (6 * pi/18) + 5 = 4 cos (6 * pi/18) + 6

6 cos (pi/3) + 5 = 4 cos (pi/3) + 6

6 (1/2) + 5 = 4 (1/2) + 6

3 + 5 = 2 + 6

8 = 8