trigonometgric equations

Question

Solve 8cos(3x)=3 for the smallest three position solutions

Answer 1

Hi Steven! Hope this helps.

1. isolate cos(3x): cos (3x) = 3/8
2. take the inverse cosine: arccos(3/8) = 3x
3. on calculator: arccos(3/8) = 68 degrees; but cosine is positive 3/8 again at 292 degrees
4. therefore: 3x = 68, 292; so x = 22.7, 97.3
5. do another revolution: 360 + 68 = 428; so 3x = 428 and x = 142.7
6. your 3 answers: 22.7, 97.3, 142.7 degrees

note: arccos is used as notation for the inverse cosine operation.

Answer 2

8cos(3x) = 3

cos(3x) = 3/8

Let θ = 3x so the problem becomes cos(θ) = 3/8 so:

θ = cos^-1(3/8)

You don't say whether you want the answer in radians or degrees, so I'll pick radians. Use your calculator to solve θ = cos^-1(3/8) and you get θ = 1.1864 radians

You will notice that the sketch of this shows that you can achieve the same cosine using θ = -1.1864:

θ = -1.1864 = 2π - 1.1864 = 5.0968

I can also go around the circle a 2nd time to get another solution of θ = 2π + 1.1864 or θ = 7.4696

We can now back substitute 3x for θ and say:

3x = 1.1864 or x = 0.3955 radians

3x = 5.0968 or x = 1.6989 radians

3x = 7.4696 or x = 2.4899 radians