Solve on the interval $[0^{\circ}, 360^{\circ})$. Hint: Factor!

$2 \sin ^2 x+3 \sin x+1=0$

Factor the given trigonometric equation 2 sin 2 x + 3 sin x + 1 = 0 by substituting y = sin x and factoring the resulting quadratic equation.
Solve for y in the factored equation ( 2 y + 1 ) ( y + 1 ) = 0 , obtaining y = − 2 1 ​ and y = − 1 .
Substitute back sin x for y and find the values of x in the interval [ 0 ∘ , 36 0 ∘ ) that satisfy sin x = − 2 1 ​ and sin x = − 1 .
The solutions are x = 21 0 ∘ , 27 0 ∘ , 33 0 ∘ , so the final answer is { 21 0 ∘ , 27 0 ∘ , 33 0 ∘ } ​ .

Explanation

Understanding the Problem We are given the trigonometric equation 2 sin 2 x + 3 sin x + 1 = 0 and asked to solve for x in the interval [ 0 ∘ , 36 0 ∘ ) . We are also given a hint to factor the equation.

Substitution Let's factor the quadratic equation. We can make a substitution y = sin x to make it easier to see the quadratic form. The equation becomes 2 y 2 + 3 y + 1 = 0 .

Factoring the Quadratic Now, we factor the quadratic equation in terms of y : ( 2 y + 1 ) ( y + 1 ) = 0

Solving for y Solve for y :
2 y + 1 = 0 or y + 1 = 0 y = − 2 1 ​ or y = − 1

Substituting Back Substitute back sin x for y :
sin x = − 2 1 ​ or sin x = − 1

Solving for sin x = -1/2 Find the values of x in the interval [ 0 ∘ , 36 0 ∘ ) for which sin x = − 2 1 ​ . The reference angle is 3 0 ∘ . Since sine is negative in the third and fourth quadrants, the solutions are: 18 0 ∘ + 3 0 ∘ = 21 0 ∘ 36 0 ∘ − 3 0 ∘ = 33 0 ∘

Solving for sin x = -1 Find the values of x in the interval [ 0 ∘ , 36 0 ∘ ) for which sin x = − 1 . This occurs at: x = 27 0 ∘

Final Solutions Therefore, the solutions are x = 21 0 ∘ , 27 0 ∘ , 33 0 ∘ .


Examples
Trigonometric equations are useful in modeling periodic phenomena such as the motion of a pendulum, the oscillations of a spring, and the propagation of light waves. Factoring trigonometric equations allows us to find specific angles at which these phenomena exhibit certain properties, such as maximum displacement or zero amplitude. For example, in electrical engineering, solving trigonometric equations helps determine the phase angles at which alternating current circuits reach peak voltage or current.

Answered by GinnyAnswer | 2025-07-03

The solutions to the equation 2 sin 2 x + 3 sin x + 1 = 0 are 21 0 ∘ , 27 0 ∘ , and 33 0 ∘ within the interval [ 0 ∘ , 36 0 ∘ ) .
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Answered by Anonymous | 2025-07-04