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.tangent, left parenthesis, theta, right parenthesis, equals, minus, one halftan(θ)=− 21
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.tangent, left parenthesis, theta, right parenthesis, equals, minus, one halftan(θ)=− 21
Solution
To solve the equation tan(θ) = -1/2 for 0 ≤ θ < 360, follow these steps:
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First, we need to find the reference angle. The reference angle is the acute angle that the terminal side of the angle makes with the x-axis. We can find it by taking the inverse tangent (or arctan) of the absolute value of the given ratio.
So, reference angle = arctan(|-1/2|) = arctan(1/2).
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Use a calculator to find the arctan(1/2). Make sure your calculator is in degree mode since we are working with degrees. You should get approximately 26.6 degrees.
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The tangent function is negative in the second and fourth quadrants. So, we subtract the reference angle from 180 degrees to find the angle in the second quadrant and subtract the reference angle from 360 degrees to find the angle in the fourth quadrant.
So, θ = 180 - 26.6 = 153.4 degrees (in the second quadrant) And, θ = 360 - 26.6 = 333.4 degrees (in the fourth quadrant)
Therefore, the solutions to the equation tan(θ) = -1/2 for 0 ≤ θ < 360 are θ = 153.4 degrees and θ = 333.4 degrees, to the nearest tenth of a degree.
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