Studdy AI Logo
Math  /  Trigonometry

QuestionUse a calculator to solve the equation on the interval 0≤θ<2π0 \leq \theta<2 \pi. 10cos⁡θ−3=010 \cos \theta-3=0 on 8 (0/1) on 12 (0/1) on 16 (0/1)
What are the solutions in the interval 0≤θ<2π0 \leq \theta<2 \pi ? Select the correct choice and fill in any answer boxes in your choice below. A. The solution set is □\square }\} (Type your answer in radians. Round to two decimal places as needed. Us a comma to separate answers as needed.) B. There is no solution.

Studdy Solution

STEP 1

1. The equation 10cos⁡θ−3=0 10 \cos \theta - 3 = 0 is a trigonometric equation.
2. We are solving for θ\theta in the interval 0≤θ<2π0 \leq \theta < 2\pi.
3. A calculator will be used to find the inverse cosine and to round the answers to two decimal places.

STEP 2

1. Isolate the cosine function.
2. Solve for θ\theta using the inverse cosine function.
3. Determine all solutions within the given interval.

STEP 3

First, isolate the cosine function by adding 3 to both sides and then dividing by 10:
10cos⁡θ−3=0 10 \cos \theta - 3 = 0 10cos⁡θ=3 10 \cos \theta = 3 cos⁡θ=310 \cos \theta = \frac{3}{10}

STEP 4

Use a calculator to find the principal value of θ\theta by taking the inverse cosine:
θ=cos⁡−1(310) \theta = \cos^{-1}\left(\frac{3}{10}\right)
Using a calculator, we find:
θ≈1.2661 radians \theta \approx 1.2661 \, \text{radians}
Round to two decimal places:
θ≈1.27 radians \theta \approx 1.27 \, \text{radians}

STEP 5

Since cosine is positive in both the first and fourth quadrants, we need to find the solution in the fourth quadrant as well. The cosine function is symmetric about the x-axis, so the second solution is:
θ=2π−cos⁡−1(310) \theta = 2\pi - \cos^{-1}\left(\frac{3}{10}\right)
Using a calculator, we find:
θ≈2π−1.2661 \theta \approx 2\pi - 1.2661 θ≈5.0171 radians \theta \approx 5.0171 \, \text{radians}
Round to two decimal places:
θ≈5.02 radians \theta \approx 5.02 \, \text{radians}

STEP 6

List all solutions within the interval 0≤θ<2π0 \leq \theta < 2\pi:
The solutions are:
θ≈1.27 radians, 5.02 radians \theta \approx 1.27 \, \text{radians}, \, 5.02 \, \text{radians}
The solution set is:
1.27,5.02 \boxed{1.27, 5.02}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord