Double & Triple Angle Formulas
Trigonometric Identities for Multiple Angles
Multiple Angle Formulas
These formulas express trigonometric functions of double (2x) and triple (3x) angles in terms of functions of the single angle (x).
They are essential for simplifying expressions, solving equations, and calculus applications.
Double Angle Formulas
Sine of 2x
\[ \sin 2x = 2 \sin x \cos x \]
\[ = \frac{2 \tan x}{1 + \tan^2 x} \]
Most Common Form: 2 sin x cos x
Cosine of 2x
\[ \cos 2x = \cos^2 x – \sin^2 x \]
\[ = 2\cos^2 x – 1 \]
\[ = 1 – 2\sin^2 x \]
\[ = \frac{1 – \tan^2 x}{1 + \tan^2 x} \]
Tangent of 2x
\[ \tan 2x = \frac{2 \tan x}{1 – \tan^2 x} \]
Note: Denominator is 1 − tan²x (not plus)
Triple Angle Formulas
Sine of 3x
\[ \sin 3x = 3 \sin x – 4 \sin^3 x \]
Pattern: 3 sin x − 4 sin³x
Cosine of 3x
\[ \cos 3x = 4 \cos^3 x – 3 \cos x \]
Pattern: 4 cos³x − 3 cos x
Tangent of 3x
\[ \tan 3x = \frac{3 \tan x – \tan^3 x}{1 – 3 \tan^2 x} \]
Similar to tan 2x: But with 3’s instead of 2’s
Key Points & Memory Tips
Cos 2x Forms
Three equivalent forms:
1. cos²x − sin²x
2. 2cos²x − 1
3. 1 − 2sin²x
Triple Angle Pattern
sin 3x: 3 sin x − 4 sin³x
cos 3x: 4 cos³x − 3 cos x
Notice the 3-4-3 pattern
When to Use
• Simplify expressions
• Solve equations
• Integration/Calculus
• Fourier series
Trigonometry Reference • Double & Triple Angle Formulas
Essential for advanced trigonometry and calculus