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Difficulty: MediumTrigonometric Graphs and Simple Equations

Find the smallest positive value of θ\theta, in degrees, that satisfies the trigonometric equation 2sin⁡(3θ−30∘)=32\sin(3\theta - 30^\circ) = \sqrt{3}.

Answer: 30 degrees

Answer

The smallest positive angle θ\theta is 30∘30^\circ.
To find the smallest positive value of θ\theta, first isolate the sine function by dividing both sides by 2 to obtain sin⁡(3θ−30∘)=32\sin(3\theta - 30^\circ) = \frac{\sqrt{3}}{2}. The smallest positive angle with a sine of 32\frac{\sqrt{3}}{2} is 60∘60^\circ. Setting 3θ−30∘=60∘3\theta - 30^\circ = 60^\circ yields 3θ=90∘3\theta = 90^\circ, which gives θ=30∘\theta = 30^\circ.

Step-by-Step Solution

1
Isolate the trigonometric ratio
sin⁡(3θ−30∘)=32\sin(3\theta - 30^\circ) = \frac{\sqrt{3}}{2}
Dividing both sides of 2sin⁡(3θ−30∘)=32\sin(3\theta - 30^\circ) = \sqrt{3} by 2 simplifies the equation into standard form.
2
Determine the primary angle solution
3θ−30∘=60∘3\theta - 30^\circ = 60^\circ
The smallest positive angle whose sine equals 32\frac{\sqrt{3}}{2} is 60∘60^\circ.
3
Solve the linear equation for θ\theta
θ=30∘\theta = 30^\circ
Adding 30∘30^\circ to both sides gives 3θ=90∘3\theta = 90^\circ, and dividing by 3 yields θ=30∘\theta = 30^\circ.

Key Concept

Solving Trigonometric Equations with Linear Argument Transformations
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Topics

Trigonometric Graphs and Simple Equations

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