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Trigonometry > Half-Angle Formulas
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QuickNotes™  

Half-Angle Formulas

Main formulas:

sin 1

2
x
=
±    /


1

2
(1− cosx)
 
cos 1

2
x
=
±    /


1

2
(1+cosx)
 

Example 1:

Use the half-angle formulas to find the sine and cosine of 15° . Check that the answers satisfy the Pythagorean identity sin2 x + cos2 x = 1.

Example 2:

Prove the following trigonometric identity:
cos 1

2
x + sin 1

2
x

cos 1

2
x − sin 1

2
x
= secx + tanx

Example 3:

Prove the following half-angle formula for tangents. Be careful about removing any ± signs!
tan 1

2
x = 1− cosx

sinx

Example 4:

Use the half-angle formulas to find the sine and cosine of (π /8). Check that the answers satisfy the Pythagorean identity sin2 x + cos2 x = 1.

Example 5:

Prove the following trigonometric identity:
tanx tan 1

2
x = secx − 1
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