Compound Angles (Trigonometry)
Introduction to Compound Angles
Compound angle formulae, also known as addition and subtraction formulae, are trigonometric identities that express trigonometric functions of sums or differences of angles (e.g., $A+B$ or $A-B$) in terms of trigonometric functions of the individual angles $A$ and $B$. These formulae are fundamental in trigonometry and have wide applications in calculus, physics, and engineering.
Compound Angle Evaluator
Compound Angle Formulae List
Sine Formulae
$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$
$$ \sin(A - B) = \sin A \cos B - \cos A \sin B $$
Cosine Formulae
$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$
$$ \cos(A - B) = \cos A \cos B + \sin A \sin B $$
Tangent Formulae
$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$
$$ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $$
Harmonic Form: $a \sin x + b \cos x$
Expressions of the form $a \sin x + b \cos x$ can be rewritten in the "harmonic form" as $R \sin(x + \alpha)$ or $R \cos(x - \alpha)$, etc. This form is useful for finding maximum/minimum values and phase shifts of trigonometric expressions.
Converting to $R \sin(x + \alpha)$
If we set $a \sin x + b \cos x = R \sin(x + \alpha)$, then by expanding $R(\sin x \cos \alpha + \cos x \sin \alpha)$, we find:
- $R = \sqrt{a^2 + b^2}$ (where $R > 0$)
- $\cos \alpha = \frac{a}{R}$ and $\sin \alpha = \frac{b}{R}$
- $\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{b/R}{a/R} = \frac{b}{a}$
The quadrant of $\alpha$ is determined by the signs of $a$ (for $\cos \alpha$) and $b$ (for $\sin \alpha$).
Harmonic Form Calculator ($R \sin(x+\alpha)$)
Convert $a \sin x + b \cos x$ to $R \sin(x + \alpha)$. Enter values for $a$ and $b$.
Double Angle Formulae Calculator
Double angle formulae express trigonometric functions of $2A$ in terms of trigonometric functions of $A$.
Formulae:
$$ \sin(2A) = 2 \sin A \cos A $$
$$ \cos(2A) = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A $$
$$ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} $$
Calculate Double Angle
Product-to-Sum Formulae Calculator
Product-to-Sum formulae allow you to rewrite products of sines and cosines as sums or differences of sines and cosines.
Formulae:
$$ 2 \sin A \cos B = \sin(A+B) + \sin(A-B) $$
$$ 2 \cos A \sin B = \sin(A+B) - \sin(A-B) $$
$$ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) $$
$$ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) $$
Calculate Product-to-Sum
Sum-to-Product Formulae Calculator
Sum-to-Product formulae allow you to rewrite sums or differences of sines and cosines as products of sines and cosines.
Formulae:
$$ \sin P + \sin Q = 2 \sin\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right) $$
$$ \sin P - \sin Q = 2 \cos\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right) $$
$$ \cos P + \cos Q = 2 \cos\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right) $$
$$ \cos P - \cos Q = -2 \sin\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right) $$