Detailed solution for cos(4x)+sin(2x)=0. for calc 1 students

1. Homework Problem Chapter 1, #43. Pg 31
Tyler Murphy
September 8, 2014
1 Solve cos 4 + cos 2 = 0
Your intuition says to

2. rst let x = 4 and y = 2.
This would give you cos x + cos y = 0:
However, this would lead to a lot of complicated reasoning. Instead,

3. rst realize that
cos 4 is really a double angle representative of cos 2. That is, cos 4 = cos 2 2. Knowing
this, let's start with the double angle formula.
Double-Angle Formula cos 2 = 2 cos2 1 (For us: cos 4 = 2 cos2 2 1)
So, using this identity we get a polynomial:
2 cos2 2 + cos 2 1 = 0
Now we factor the left-hand side:
(1 + cos 2)(2 cos 2 1) = 0
Now we solve each of the binomials independently.
1 + cos 2 = 0 (1)
and
2 cos 2 1 = 0 (2)
Let's

4. rst work on equation (1).
1 + cos (2) = 0
cos (2) = -1
cos1 (2) = cos1 (1)
2 = + 2 k1 for some integer k1
=
2
+ k1
1

5. Now let's look at equation (2).
2 cos 2 1 = 0
2 cos 2 = 1
cos 2 =
1
2
cos1 (cos 2) = cos1 (
1
2
)
2 =
3
+ 2k2 or
5
3
+ 2k3 for some integers k2 and k3.
=
6
+ k2 or
5
6
+ k3
So our

6. nal solutions are:
=
2
+ k1
=
6
+ k2
=
5
6
+ k3
Now you simply need to plug in values for k1; k2; k3 to

7. nd concrete values for in your
given range.
2