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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

By squaring both sides of the equations we can obtain

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

What is the exact value of cos(15º)?
What is your answer?
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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