Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees, comma0 ∘ ≤θ<360 ∘ , that satisfy the equation below, to the nearest tenth of a degree.sine, left parenthesis, theta, right parenthesis, equals, minus, start fraction, square root of, 3, end square root, divided by, 8, end fractionsin(θ)=− 83
Question
Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees, comma0 ∘ ≤θ<360 ∘ , that satisfy the equation below, to the nearest tenth of a degree.sine, left parenthesis, theta, right parenthesis, equals, minus, start fraction, square root of, 3, end square root, divided by, 8, end fractionsin(θ)=− 83
Solution
The given equation is sin(θ) = -√3/8.
Step 1: Find the reference angle We know that sin(θ) = y/r, where y is the opposite side and r is the hypotenuse in a right triangle. Here, y = -√3 and r = 8. The negative sign indicates that the angle is in the third or fourth quadrant.
Step 2: Use the inverse sine function to find the reference angle θ = sin^(-1)(√3/8) ≈ 20.9 degrees
Step 3: Find the actual angles Since the sine is negative, the angle is in the third or fourth quadrant. So, the actual angles are: θ = 180 + 20.9 = 200.9 degrees (third quadrant) θ = 360 - 20.9 = 339.1 degrees (fourth quadrant)
So, the angles that satisfy the equation sin(θ) = -√3/8 in the interval 0 ≤ θ < 360 are 200.9 degrees and 339.1 degrees, to the nearest tenth of a degree.
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