# Trigonometry : Finding Trigonometric Roots

## Example Questions

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### Example Question #1 : Finding Trigonometric Roots

Which of the following is a solution to the following equation such that

Explanation:

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.

### Example Question #1 : Finding Trigonometric Roots

Solve the following equation for .

No solution exists

Explanation:

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

### Example Question #1 : Finding Trigonometric Roots

Solve the equation for .

No solution exists

Explanation:

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

### Example Question #1 : Finding Trigonometric Roots

Solve the following equation for .

Explanation:

;  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 .

### Example Question #1 : Finding Trigonometric Roots

Solve the following equation for .

No solution exists

Explanation:

;  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

### Example Question #1 : Finding Trigonometric Roots

Solve the equation for .

Explanation:

;  Divide both sides by 3

;  Take the square root on both sides.  Just as the previous question, when you take a square root the answer could be positive or negative.

;  This can be written as two separate equations

and  ;  Take the inverse tangent

and

### Example Question #7 : Finding Trigonometric Roots

Solve the following equation for .

Explanation:

;  The expression is similar to a quadratic expression and can be factored.

;  set both expressions equal to 0.  Since they are the same, the solutions will repeat, so I will only write it once.

;  take the inverse tangent on both sides

### Example Question #1 : Finding Trigonometric Roots

Solve the equation below for  greater than or equal to  and strictly less than

and

and

only

only

and

Explanation:

Recall the values of  for which . If it helps, think of sine as the  values on the unit circle. Thus, the acceptable values of  would be 0, 180, 360, 540 etc.. However, in our scenario .

Thus we have  and .

Any other answer would give us values greater than 90. When we divide by 4, we get our answers,

and

### Example Question #2 : Finding Trigonometric Roots

Solve the following equation.  Find all solutions such that .

No Solution

Explanation:

; Divide both sides by 2 to get

; take the inverse sine on both sides

; the left side reduces to x, so

At this point, either use a unit circle diagram or a calculator to find the value.

Keep in mind that the problem asks for all solutions between  and .

If you use a calculator, you will only get as an answer.

So we need to find another angle that satisfies the equation .

### Example Question #11 : Solving Trigonometric Equations

Solve the following equation.  Find all solutions such that .

No solution exists

Explanation:

;  First use the double angle identity for .

;  divide both sides by 2

;  subtract the from both sides

;  factor out the

; Now we have the product of two expressions is 0.  This can only happen if one (or both) expressions are equal to 0.  So let each expression equal 0.

or

or  ;  Take the inverse of each function for each expression.

or ;  The second equation is not possible so gives no solution, but the first equation gives us:

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