Set the two equations equal to each other

subtract cos from both sides

take the square root of both sides

Set both equations equal to each other:

subtract from both sides

subtract from both sides

We can re-write the left side using a trigonometric identity

take the inverse cosine

divide by 2

Set the two equations equal to each other:

subtract from both sides

add 5 to both sides

divide both sides by 4

take the square root of both sides

A number x is a solution if it satisfies both equations.

We note first we can write the first equation in the form :

We know that for all reals. This means that there is no x that

satisifies the first inequality. This shows that the system cannot satisfy both equations since it does not satisfy one of them. This shows that our system does not have a solution.

First, set both equations equal to each other:

 subtract from both sides

Using a trigonometric identity, we can re-write as :

 combine like terms

subtract 2 from both sides

We can solve for using the quadratic formula:

This gives us 2 possible values for cosine

We begin by getting the right side of the equation to equal zero.

Next we factor.

We then set each factor equal to zero and solve.

            or        

We then determine the angles that satisfy each solution within one revolution.

The angles  and  satisfy the first, and  satisfies the second.  Only  is among our answer choices.

The fastest way to solve this problem is to substitute a new variable.  Let .

The equation now becomes:

So at what angles are the sine and cosine functions equal.  This occurs at

You may be wondering, "Why did you include

if they're not between and ?"

The reason is because once we substitute back the original variable, we will have to divide by 2.  This dividing by 2 will bring the last two answers within our range.

Dividing each answer by 2 gives us

We begin by substituting a new variable .

;  Use the double angle identity for .

;  subtract the from both sides.

;  This expression can be factored.

;  set each expression equal to 0.

or ;  solve each equation for

or ;  Since we sustituted a new variable we can see that if , then we must have .  Since , that means .

This is important information because it tells us that when we solve both equations for u, our answers can go all the way up to not just .

So we get

  Divide everything by 2 to get our final solutions

;  We start by substituting a new variable.  Let .

; Use the double angle identity for cosine

;  the 1's cancel, so add to both sides

;  factor out a from both terms.

;  set each expression equal to 0.

  or  ;  solve the second equation for sin u.

  or  ;  take the inverse sine to solve for u (use a unit circle diagram or a calculator)

;  multiply everything by 2 to solve for x.

;  Notice that the last two solutions are not within our range  .  So the only solution is .

;  First divide both sides of the equation by 4

;  Next take the square root on both sides.  Be careful. Remember that when YOU take a square root to solve an equation, the answer could be positive or negative.  (If the square root was already a part of the equation, it usually only requires the positive square root.  For example, the solutions to are 2 and -2, but if we plug in 4 into the function the answer is only 2.)  So,

;  we can separate this into two equations

  and  ;  we get

and