Solve for all values of thetaθ, such that 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding all values to the nearest tenth.2, cotangent, theta, plus, 1, equals, 02cotθ+1=0
Question
Solve for all values of thetaθ, such that 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding all values to the nearest tenth.2, cotangent, theta, plus, 1, equals, 02cotθ+1=0
Solution
To solve the equation 2cotθ + 1 = 0 for all values of θ such that 0 ≤ θ < 360, follow these steps:
Step 1: Isolate cotθ 2cotθ = -1 cotθ = -1/2
Step 2: Find the values of θ The cotangent function is negative in the second and fourth quadrants.
In the second quadrant, θ = 180 - α, where α is the angle whose cotangent is 1/2. In the fourth quadrant, θ = 360 - α.
Step 3: Find α We know that cotα = 1/tanα. So, tanα = -2 (since cotθ = -1/2).
Using a calculator, we find that α ≈ 63.4 degrees.
Step 4: Substitute α into the equations from Step 2 In the second quadrant, θ = 180 - 63.4 = 116.6 degrees. In the fourth quadrant, θ = 360 - 63.4 = 296.6 degrees.
So, the solutions to the equation 2cotθ + 1 = 0 are θ = 116.6 degrees and θ = 296.6 degrees.
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