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Chapter Contents

• Analytic Trigonometry
• 1. Trigonometric Identities
• 2. Sin, cos tan of Sum of Two Angles
• 3. Double Angle Formulas
• 4. Half Angle Formulas
• 5. Trigonometric Equations
• 6. Express in the form R sin (θ + α)
• 7. Inverse Trigonometric Functions

• Comments, Questions?

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4. Half-Angle Formulas

by M. Bourne

We will develop formulas for the sine, cosine and tangent of a half angle.

Half Angle Formula - Sine

We start with the formula for the cosine of a double angle that we met in the last section.

cos 2θ = 1− 2sin2 θ

Now, if we let

then 2θ = α and our formula becomes:

cos α = 1 − 2sin² (α/2)

We now solve for

(That is, we get sin(α/2) on the left of the equation and everything else on the right):

2sin²(α/2) = 1 − cos α

sin²(α/2) = (1 − cos α)/2

Solving gives us the following sine of a half-angle identity:

The sign of depends on the quadrant in which α/2 lies.

If α/2 is in the first or second quadrants, the formula uses the positive case:

If α/2 is in the third or fourth quadrants, the formula uses the negative case:

Half Angle Formula - Cosine

Using a similar process, with the same substitution of (so 2θ = α) we subsitute into the identity

cos 2θ = 2cos² θ − 1 (see cosine of a double angle)

We obtain

Reverse the equation:

Add 1 to both sides:

Divide both sides by 2

Solving for cos(α/2), we obtain:

As before, the sign we need depends on the quadrant.

If α/2 is in the first or fourth quadrants, the formula uses the positive case:

If α/2 is in the second or third quadrants, the formula uses the negative case:

Half Angle Formula - Tangent

We can show the tangent of a half angle is given by:

Proof

We can show this is equivalent to:

Proof

Summary of Tan of a Half Angle

Using t

It is sometimes useful to define t as the tan of a half angle:

This gives us the results:

Tan of the Average of 2 Angles

With some algebraic manipulation, we can obtain:

Example 1

Find the value of sin 15° using the sine half-angle relationship given above.

Answer

Example 2

Find the value of cos 165° using the cosine half-angle relationship given above.

Answer

Example 3:

Show that

Answer

Exercises: Evaluating and Proving Half-Angle Identities

1. Use the half angle formula to evaluate sin 75°.

Answer

2. Find the value of if where 0° < α < 90°.

Answer

3. Prove the identity:

Answer

4. Prove the identity:

Answer