In Exercises 97–116, use the most appropriate method to solve ...

Question

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos x - 5 = 3 cos x + 6

Accepted Answer

Welcome back. I am so glad you're here. We're asked to apply the most suitable technique to determine the solution of the following equation within the interval from 0 to 2 pi inclusive of zero but not of two pi employing exact values whenever feasible or providing approximations accurate to four significant figures. Our equation is two cosine A minus 19 equals five cosine A plus 11. Our answer choices are answer choice A XX equals the solution set zero pi two pi answer choice B X equals the solution set pi divided by four and seven pi divided by four. Answer choice C X equals the solution set pi divided by 12 and five pi divided by 12 and answer choice D no solution. All right. Looking at our equation, I want to get all of the cosines on one side of the equal sign and all of the constant terms on the other side. So I am going to add to both sides of the equation and I am going to subtract five cosine A from both sides of the equation. So on the left two cosine A minus five cosine A will leave us with negative three cosine A and the negative plus 19 is zero. So that equals on the right five cosine A minus five cosine A is zero and 11 plus 19 is a positive 30. Now dividing both sides by negative three to get the cosine A by itself on the left, we have cosine A and that equals on the right 30 divided by negative three is negative 10. So we have the cosine of angle A equals negative 10. So what we can do is we can find the solution for angle A by finding the inverse cosine of what's on the other side, which is negative 10. So the inverse cosine of negative 10. However, and be sure that you have your calculator set to radiance here. If you input the negative code or excuse me, not negative the inverse cosine of negative 10, you're going to get a domain error. And that's because the domain for inverse cosine is from negative one to positive one and negative 10 is beyond that domain. So there is actually no solution for this problem which matches with answer choice D well done. We'll catch you on the next one.

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos x - 5 = 3 cos x + 6

Key Concepts

Solving Trigonometric Equations

Interval Restriction [0, 2π)

Using Exact and Approximate Values

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