Question

Solve, for 0≤x≤2, the equation 1 + 5cos 3x = 0, giving your answer in radians correct to 2 decimal places.

Answer 1

\[
1 + 5\cos(3x) = 0 \\
5\cos(3x) = -1 \\
\cos(3x) = -0.2 \\
3x = \cos^{-1}(-0.2) \\
\]Find the inverse cosine of -0.2 using your calculator. List all values from 0 to 6 radians. (because 0≤ x ≤2, 0≤ 3x ≤ 6).

Answer 2

i know how to do it, but the mark scheme is making me confuse. can you show all of your working out proccess? ;; A ;;

Answer 3

Set the calculator to radian mode and find inverse cosine of -0.2. We get 1.7722 radians. If the reference angle is theta, cosine has the same value for theta and (2pi - theta) radians.

Answer 4

So \(1.7722\) radians is one solution and \((2\pi - 1.7722)\) is another solution.
\(1.7722\) and \(4.5110\) radians. And since cosine function has a period of \(2\pi\) or \(6.2832\) radians we can add multiples of that number to the above as long as it does not exceed 6 because \(x \le 2\) which implies \(3x \le 6\). Here we cannot add anymore multiples of \(2\pi\).
So \(3x = 1.7722, 4.5110\).
\(x = 0.5907, 1.5037\).
Round it to two decimal places: \(x = 0.59,~1.50 \text{ radians.}\)

Answer 5

i thought the 2 angles should be in the opposite quadrants since the cos is negative?

Answer 6

Correct. For this problem, it will be in the second and third quadrants. But the above drawing is to show that cos(theta) is same as cos(2pi - theta) which is true no matter which quadrant theta is in.

Answer 7

so will the answer be in the first and fourth quadrants, i'm still quite unsure. since the mark scheme said that it should be in the first and fourth quadrants, will i think it should be in the opposite

Answer 8

90 degrees is pi/2 radians which is 1.57 radians.
we got x = 0.59 and 1.5 radians both of which are in the first quadrant.

But we we solving for cos(3x) = -0.2
cosine is negative in the second and third quadrant. So when we take the inverse cosine of -0.2, we will get angles in the second and third quadrant. They were 1.7722 radians (101.5 degrees, second quadrant) and 4.5110 radians (258.5 degrees, third quadrant).

Answer 9

Then we had to divide those two angles by 3 to get x.