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Half and Double Angle Formulas
Introduction: In this lesson, formulas involving half of and twice of an angle will be defined and applied to the fundamental trig functions.

The Lesson:
For any angle a we have the following relationships:
Half angle formulas:

Double angle formulas:
We will use these formulas to determine the exact values of trig functions of certain angles in terms of half or double values. Proofs are available in all trig and pre-calculus texts.

Two other formulas can be derived from
and

.
By squaring both sides of the equations we can obtain
and

 
If we let A = we have
.

Let's Practice:
  1. What is the exact value of tan(15º)?
We can use a half angle formula noticing that .
We have tan(15º) = tan() =
.
  1. A quadrant four angle A has a tangent of .
    What is the exact value of sin(2A)?
In the diagram of angle A shown below, the hypotenuse would be .

To find the sin(2A) we use the double angle formula
  1. Find the and the for the angle in example (ii).
To find the we use the half angle formula
.
Since angle A is in quadrant four, we have . Dividing by 2 gives us which puts angle in quadrant two. Therefore the sine is positive and
To find the we use the half angle formula
.

Examples
Example
What is the exact value of cos(15º)?
What is your answer?
 
Example
A quadrant three angle a has a cosine of -0.9. What is the exact value of tan(2a)?
What is your answer?
 



M Ransom

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