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 thirdtan(θ)=− 31
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 thirdtan(θ)=− 31
Solution
To solve for θ in the equation tan(θ) = -1/3, we need to find the angle whose tangent is -1/3.
Step 1: Use the inverse tangent function (also known as arctan or tan^-1) to find the reference angle.
θ = arctan(-1/3)
This will give you the angle in the 4th quadrant because tangent is negative in the 4th quadrant.
Step 2: Convert this angle to a positive acute angle by adding 180 degrees if it's negative.
Step 3: Since tangent has a period of 180 degrees, add 180 degrees to the result from step 2 to find the angle in the 2nd quadrant.
Step 4: Check that both angles are within the given range 0 ≤ θ < 360.
Remember to round your answers to the nearest tenth of a degree.
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