# Trigonometric Identities and Equations

## Double-Angle, Half-Angle, and Reduction Formulas

Bicycle ramps made for competition (see ) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be 
     such that 
     The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.

### Using Double-Angle Formulas to Find Exact Values

In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where 
       Deriving the double-angle formula for sine begins with the sum formula,
If we let 
       then we have
Deriving the double-angle for cosine gives us three options. First, starting from the sum formula, 
       and letting 
       we have
Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. The first one is:
The second interpretation is:
Similarly, to derive the double-angle formula for tangent, replacing 
       in the sum formula gives

### Using Double-Angle Formulas to Verify Identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

### Use Reduction Formulas to Simplify an Expression

The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.
We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with 
       Solve for 
      
    
Next, we use the formula 
       Solve for 
      
    
The last reduction formula is derived by writing tangent in terms of sine and cosine:

###  Using Half-Angle Formulas to Find Exact Values 

The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace 
       with 
       the half-angle formula for sine is found by simplifying the equation and solving for 
       Note that the half-angle formulas are preceded by a 
       sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which 
       terminates.
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have

### Key Equations


### Key Concepts

1. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See , ,  , and .
2. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See  and .
3. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See , , and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the exact values of a) 
         b) 
         and c) 
         without solving for 
        
      
For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.
For the following exercises, simplify to one trigonometric expression.
For the following exercises, find the exact value using half-angle formulas.
For the following exercises, find the exact values of a) 
         b) 
         and c) 
         without solving for 
        
      when 
         
For the following exercises, use  to find the requested half and double angles.
For the following exercises, simplify each expression. Do not evaluate.
For the following exercises, prove the identity given.
For the following exercises, rewrite the expression with an exponent no higher than 1.

### Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.
For the following exercises, algebraically find an equivalent function, only in terms of 
         and/or 
         and then check the answer by graphing both equations.

### Extensions

For the following exercises, prove the identities.