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Math  /  Trigonometry

QuestionSolve the equation. Give all solution in 0≤θ<2π0 \leq \theta<2 \pi 2cos⁡(θ)=12 \cos (\theta)=1
Select all that apply 0 π2\frac{\pi}{2} π\pi 3π2\frac{3 \pi}{2} −π6-\frac{\pi}{6} π6\frac{\pi}{6} 2π3\frac{2 \pi}{3} 7π6\frac{7 \pi}{6} 5π3\frac{5 \pi}{3} −π4-\frac{\pi}{4} π4\frac{\pi}{4} 3π4\frac{3 \pi}{4} 5π4\frac{5 \pi}{4} 7π4\frac{7 \pi}{4} −π3-\frac{\pi}{3} π3\frac{\pi}{3} 5π6\frac{5 \pi}{6} 4π3\frac{4 \pi}{3} 11π6\frac{11 \pi}{6} −π2-\frac{\pi}{2}

Studdy Solution

STEP 1

1. The equation 2cos⁡(θ)=1 2 \cos(\theta) = 1 is a trigonometric equation.
2. We are looking for solutions within the interval 0≤θ<2π 0 \leq \theta < 2\pi .

STEP 2

1. Isolate the cosine function.
2. Determine the angle(s) that satisfy the equation.
3. Verify the solutions within the given interval.

STEP 3

First, isolate the cosine function by dividing both sides of the equation by 2:
2cos⁡(θ)=1 2 \cos(\theta) = 1 cos⁡(θ)=12 \cos(\theta) = \frac{1}{2}

STEP 4

Determine the angles that have a cosine value of 12\frac{1}{2}. The cosine function equals 12\frac{1}{2} at specific angles in the unit circle. These angles are:
θ=π3andθ=5π3 \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{5\pi}{3}

STEP 5

Verify that these solutions are within the interval 0≤θ<2π 0 \leq \theta < 2\pi .
- π3\frac{\pi}{3} is within the interval 0≤θ<2π 0 \leq \theta < 2\pi . - 5π3\frac{5\pi}{3} is also within the interval 0≤θ<2π 0 \leq \theta < 2\pi .
The solutions to the equation 2cos⁡(θ)=1 2 \cos(\theta) = 1 in the interval 0≤θ<2π 0 \leq \theta < 2\pi are:
π3,5π3 \boxed{\frac{\pi}{3}, \frac{5\pi}{3}}

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