How many values of in the interval 0< 2 satisfy 1-3 +5 3 = 0?

We rearrange to get 5 3 = 3 -1. We can graph two functions in this case: y=5 3x and y=3 x -1 . Using transformation of functions, we know that 5 3x is just a cosine function with amplitude 5 and period 2 3 . Similarly, 3 x -1 is just a sine function with amplitude 3 and shifted 1 unit downward: So, we have (D) 6 solutions.

Trigonometric Equations Difficulty 4

TMUA practice: Trig Equations, problem 2

A TMUA-calibre trig equations problem at difficulty 4 of 5, with the full worked solution.

The question

How many values of

\theta

in the interval

0<\theta\le 2\pi

satisfy

1-3\sin\theta+5\cos3\theta = 0?

2

4

5

6

8

The correct answer is highlighted. D

Worked solution

We rearrange to get

$5\cos3\theta = 3\sin\theta-1.$

We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$ . Using transformation of functions, we know that $5\cos{3x}$ is just a cosine function with amplitude $5$ and period $\frac{2\pi}{3}$ . Similarly, $3\sin{x} -1$ is just a sine function with amplitude $3$ and shifted $1$ unit downward:

So, we have $\boxed{\textbf{(D) }6}$ solutions.