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Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , that solve the following equation.cosine, theta, equals, minus, start fraction, square root of, 3, end square root, divided by, 2, end fractioncosθ=− 23​ ​

Question

Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , that solve the following equation.cosine, theta, equals, minus, start fraction, square root of, 3, end square root, divided by, 2, end fractioncosθ=− 23​ ​

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Solution

The given equation is cosθ = -√3/2.

The cosine function is negative in the second and third quadrants of the unit circle.

In the first step, we find the reference angle. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. The cosine of 30 degrees or π/6 radians is √3/2. So, the reference angle is 30 degrees.

In the second quadrant, the angle is found by subtracting the reference angle from 180 degrees. So, θ = 180 - 30 = 150 degrees.

In the third quadrant, the angle is found by adding the reference angle to 180 degrees. So, θ = 180 + 30 = 210 degrees.

So, the solutions to the equation cosθ = -√3/2 in the interval 0 ≤ θ < 360 are θ = 150 degrees and θ = 210 degrees.

This problem has been solved

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