Question

find all solutions to the following equation:
7 sin θ + 1 = 5 sin θ
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. θ = □
(simplify your answer. type your answer(s) as an expression, using n as the variable, in the form a + bn where 0 ≤ a < 2π. type any angle measures in radians, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)
b. there is no solution.

Explanation:

Step1: Simplify the equation

Subtract \(5\sin\theta\) from both sides:
\(7\sin\theta + 1-5\sin\theta=5\sin\theta - 5\sin\theta\)
\(2\sin\theta+1 = 0\)
Then subtract \(1\) from both sides:
\(2\sin\theta=- 1\)
Divide both sides by \(2\):
\(\sin\theta=-\frac{1}{2}\)

Step2: Find the general solution

We know that \(\sin\theta =-\frac{1}{2}\) when \(\theta=\frac{7\pi}{6}+2n\pi\) or \(\theta=\frac{11\pi}{6}+2n\pi\), where \(n\in\mathbb{Z}\)

Answer:

\(\theta=\frac{7\pi}{6}+2n\pi,\frac{11\pi}{6}+2n\pi\)