Chapter 10 - NCPN

Lesson 13.4 Double-Angle and
Half-Angle Identities
Objectives
Find exact values of
trigonometric expressions
using double-angle and
half-angle identities.
A radio wave is an electromagnetic wave
transmitted by an antenna. Radio waves
have different frequencies. When tuning
a radio receiver to a specific frequency, a
specific signal can be picked up. Listening
to a radio station, such as 99.1 FM,
“The Wave,” means that a radio station
is broadcasting an FM radio signal at a
frequency of 99.1 megahertz. Megahertz
means “millions of cycles per second.”
So “99.1 megahertz” means that the
transmitter at the radio station is oscillating
at a frequency of 99,100,000 cycles per
second. A transmitter at the radio station
needs to evaluate sin 150° to isolate a radio
frequency. What is the isolated frequency
Double-Angle Identities
The Double Angle Identities are special cases of the Angle Sum
Identities in which A = B. If A = B, then the cos (A + A) can be
expressed as cos 2A.
Therefore, cos (A + A) = cos A cos A – sin A sin A
= cos 2 A – sin 2 A
Note that the Double-Angle Identity for cos has multiple variations.
These variations can be derived using trigonometric identities.
The table below summarizes the Double-Angle Identities.
Double-Angle Identities
• cos 2θ = cos 2 θ – sin 2 θ
• cos 2θ = 2cos 2 θ – 1
• cos 2θ = 1 – 2sin 2 θ
• sin 2θ = 2sin θ cos θ
• tan 2θ =
2tan
θ
2
1−
tan θ
10.8 Double-Angle and Half-Angle Identities 479