Math  /  Trigonometry

Question13. If cosθ=0.5000\cos \theta=0.5000, find the value of θ\theta correct to two decimal places, where 0θ<2π0 \leq \theta<2 \pi.
13a Find the acute angle θ\theta that solves the equation. θ=\theta= Enter your next step here

Studdy Solution

STEP 1

1. We are given that cosθ=0.5000\cos \theta = 0.5000.
2. We need to find θ\theta in radians within the interval 0θ<2π0 \leq \theta < 2\pi.
3. We will use the inverse cosine function to find the principal value of θ\theta.

STEP 2

1. Use the inverse cosine function to find the principal angle.
2. Verify and adjust the angle to ensure it is within the specified interval.
3. Determine if there are other angles that satisfy the equation within the interval.

STEP 3

Use the inverse cosine function to find the principal angle θ\theta:
θ=cos1(0.5000) \theta = \cos^{-1}(0.5000)
The principal value of θ\theta is:
θ=π3 \theta = \frac{\pi}{3}
Convert θ\theta to a decimal value:
θ1.05radians \theta \approx 1.05 \, \text{radians}

STEP 4

Since cosine is positive in the first and fourth quadrants, we need to check for any additional solutions in the interval 0θ<2π0 \leq \theta < 2\pi.
The second solution in the fourth quadrant is given by:
θ=2ππ3 \theta = 2\pi - \frac{\pi}{3}
Calculate this value:
θ=6π3π3=5π3 \theta = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
Convert θ\theta to a decimal value:
θ5.24radians \theta \approx 5.24 \, \text{radians}

STEP 5

Verify both solutions are within the specified interval 0θ<2π0 \leq \theta < 2\pi.
Both θ1.05\theta \approx 1.05 and θ5.24\theta \approx 5.24 are within the interval.
The values of θ\theta correct to two decimal places are:
θ1.05radiansandθ5.24radians \theta \approx 1.05 \, \text{radians} \quad \text{and} \quad \theta \approx 5.24 \, \text{radians}

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