Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , that solve the following equation.sine, theta, equals, minus, start fraction, square root of, 2, end square root, divided by, 2, end fractionsinθ=− 22
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.sine, theta, equals, minus, start fraction, square root of, 2, end square root, divided by, 2, end fractionsinθ=− 22
Solution
The given equation is sinθ = -√2/2.
Step 1: We know that sine is negative in the third and fourth quadrants.
Step 2: We also know that sin(45°) = √2/2. Therefore, sin(-45°) = -√2/2.
Step 3: To find the angles in the third and fourth quadrants, we add and subtract 180° from -45° respectively.
So, θ = 180° - 45° = 135° (in the third quadrant) and θ = 360° - 45° = 315° (in the fourth quadrant).
Therefore, the solutions to the equation sinθ = -√2/2 in the interval 0° ≤ θ < 360° are θ = 135° and θ = 315°.
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