Find to the nearest degree, all values of ө less than 360˚ that satisfy the equation 2tan^2θ-2tanθ=3

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solve 2tan^2è-2tanè=3

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Solutions are :

è = -58.71608 °

è = - 59 ° to the nearest degree

and

è = 61.2515 °

è = 61 ° to the nearest degree

è

is greek letter theta

To find the values of θ that satisfy the equation, we can follow these steps:

Step 1: Rewrite the equation using the identity tan^2θ = 1 + sec^2θ.
The equation becomes: 2(1 + sec^2θ) - 2tanθ = 3.

Step 2: Simplify the equation.
2 + 2sec^2θ - 2tanθ = 3.
2sec^2θ - 2tanθ = 1.

Step 3: Divide the entire equation by 2.
sec^2θ - tanθ = 1/2.

Step 4: Re-write the equation in terms of sine and cosine.
Using the identity sec^2θ = 1 + tan^2θ, the equation becomes:
(1 + tan^2θ) - tanθ = 1/2.
tan^2θ - tanθ - 1/2 = 0.

Step 5: Solve the quadratic equation.
We can solve this quadratic equation by factoring or using the quadratic formula. Factoring may not lead to exact solutions, so we'll use the quadratic formula.
tanθ = [1 ± √(1 - 4 * 1 * (-1/2))] / (2 * 1)
tanθ = [1 ± √(1 + 2)] / 2
tanθ = [1 ± √3] / 2.

Step 6: Solve for θ.
We need to find the values of θ for which tanθ = [1 ± √3] / 2. The tangent function repeats every 180 degrees, so we can consider the two possible angles: θ1 and θ2.

θ1 = tan^(-1)((1 + √3) / 2)
θ2 = tan^(-1)((1 - √3) / 2)

Using a calculator, we find:
θ1 ≈ 56.3 degrees
θ2 ≈ 246.3 degrees

Therefore, the values of θ less than 360 degrees that satisfy the equation 2tan^2θ - 2tanθ = 3 are approximately 56.3 degrees and 246.3 degrees.