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Explain Bot To solve the equation 4cos(θ)tan(θ) + 15 = 0 for 0 ≤ θ ≤ 360, we need to find the values of θ that satisfy the equation. Here's how you can do it:
Step 1:
Let's simplify the equation by multiplying through by the common denominator of cos(θ) and tan(θ), which is sin(θ), to eliminate the denominators. This gives us:
4cos(θ)sin(θ) + 15sin(θ) = 0
Step 2:
Using the trigonometric identity sin(2θ) = 2cos(θ)sin(θ), we can rewrite the equation as:
2sin(2θ) + 15sin(θ) = 0
Step 3:
Now, let's factor out sin(θ) from the equation:
sin(θ) (2cos(θ) + 15) = 0
Step 4:
Now we have two possibilities for sin(θ) that satisfy the equation:
1. sin(θ) = 0
2. 2cos(θ) + 15 = 0
For the first possibility, sin(θ) = 0, the solutions are θ = 0° and θ = 180°.
For the second possibility, 2cos(θ) + 15 = 0, we need to solve for θ. Let's isolate cos(θ):
2cos(θ) = -15
cos(θ) = -15/2
Now, using the inverse trigonometric function, we can find the values of θ for which cos(θ) = -15/2. Note that COS^-1 in some calculators is represented as arccos or cos^(-1).
θ = COS^-1(-15/2)
Using a calculator, the approximate values of θ are:
θ ≈ 113.58°
θ ≈ 246.42°
Therefore, the solutions to the equation 4cos(θ)tan(θ) + 15 = 0 for 0 ≤ θ ≤ 360 are:
θ = 0°, θ = 180°, θ ≈ 113.58°, and θ ≈ 246.42°.