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 sixthtan(θ)=− 61
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 sixthtan(θ)=− 61
Solution
To solve the equation tan(θ) = -1/6 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 arctangent of the absolute value of the ratio. In this case, the ratio is -1/6. So, the reference angle is arctan(|-1/6|) = arctan(1/6).
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Use a calculator to find the arctangent of 1/6. Make sure your calculator is in degree mode. You should get approximately 9.5 degrees.
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The tangent function is negative in the second and fourth quadrants. So, we need to find the angles in these quadrants that have the same reference angle.
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In the second quadrant, the angle is 180 - 9.5 = 170.5 degrees.
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In the fourth quadrant, the angle is 360 - 9.5 = 350.5 degrees.
So, the solutions to the equation tan(θ) = -1/6 for 0 ≤ θ < 360 are θ = 170.5 degrees and θ = 350.5 degrees.
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