Chapter 10 - NCPN

Example 1
Using the Double-Angle Identity
Use a double-angle identity to find the exact value of cos 240°.
Solution
Because sin 120° = 3
2 , the identity cos 2θ = 1 – 2 sin2 θ can be used to
find the exact value of cos 240°.
cos 2θ = 1 – 2 sin 2 θ
cos (2 • 120°) = 1 – 2(sin 120°) 2
⎛
cos 240° = 1 – 2⎜
⎝
cos 240° = 1 – 2 3 4
cos 240° = − 1 2
3
2
( )
⎞
⎟
⎠
The exact value of cos 240° is − 1 2 .
Ongoing Assessment
2
Use a double-angle identity to find the exact value of cos 120°. − 1 2
Half-Angle Identities
The table below summarizes the Half-Angle Identities. The Half Angle
Identities can be derived from the Double Angle Identities. As part of
the derivation process, you must take the square root of both sides of the
equation. Recall that you must include ± when taking the square root of both
sides of an equation.
Half-Angle Identities
• sin
A
=±
2
• cos
A
=±
2
• tan
A
=±
2
1−
cos A
2
1+
cos A
2
1−
cos A
1+
cos A
When using the half-angle identities, choose the sign for each function as
appropriate for the angle. For example, the sine function is positive if the
terminal side of the angle lies in quadrants I or II. Otherwise, it is negative.
The table at the top of the next page summarizes this concept.
480 Chapter 10 Trigonometric Functions and Identities