Reciprocal Identities

Quotient Identities

Even Functions (Symmetric about the y-axis)

Odd Functions (Symmetric about the origin)

Co-function identities show the relationship between a trigonometric function and its co-function evaluated at the complementary angle.

• $\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$
• $\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$
• $\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta$
• $\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta$
• $\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta$
• $\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta$

Primary Identity:

Derived Identities: Divide by $\cos^2 \theta$:

Divide by $\sin^2 \theta$:

• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
• $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

• $\sin(2\theta) = 2\sin \theta \cos \theta$
• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
• $\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}$

• $\sin(3\theta) = 3\sin \theta - 4\sin^3 \theta$
• $\cos(3\theta) = 4\cos^3 \theta - 3\cos \theta$
• $\tan(3\theta) = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}$

These are derived by rearranging the double angle formulas for cosine. They are extremely important in calculus for integrating even powers of sine and cosine.

• $\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$
• $\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$
• $\tan^2 \theta = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$

• $\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$
• $\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
• $\tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}$

Note: The sign ($\pm$) depends on the quadrant of the half-angle $\theta/2$.

Before the invention of logarithms in the early 17th century, multiplying large numbers was extremely time-consuming and error-prone. Mathematicians and astronomers used a technique called Prosthaphaeresis (from Greek, meaning "addition and subtraction").

They used the Product-to-Sum formulas backward:

To multiply two numbers, they would scale them to be between 0 and 1, look up the angles whose cosines matched those numbers in a trig table, add and subtract the angles, find the cosines of those new angles, and simply average them. This reduced multiplication to simple addition and subtraction!

A linear combination of a sine and cosine wave of the same frequency can be written as a single sine or cosine wave with a phase shift. This is extremely useful in alternating current (AC) circuit analysis and wave mechanics.

Where:

• $R = \sqrt{A^2 + B^2}$ (the new amplitude)
• $\tan \alpha = \frac{B}{A}$ (the phase shift angle, ensure $\alpha$ is in the correct quadrant based on the signs of $A$ and $B$)

Alternatively, it can be expressed in terms of cosine:

Where $\tan \beta = \frac{A}{B}$.

The tangent half-angle substitution, also known as the Weierstrass substitution, is a powerful technique for converting trigonometric expressions into rational algebraic expressions. It is primarily used to evaluate complex trigonometric integrals in calculus.

Let $t = \tan\left(\frac{\theta}{2}\right)$. The fundamental trigonometric functions can then be expressed entirely in terms of $t$:

By substituting these algebraic equivalents, a difficult trigonometric problem is often reduced to a manageable algebraic one.

In advanced mathematics and engineering, memorizing all trigonometric identities is often unnecessary due to Euler's Formula:

From this, sine and cosine can be expressed algebraically in terms of complex exponentials:

• $\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$
• $\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

By substituting these exponential definitions, any trigonometric identity (like Angle Sum or Double Angle) can be derived purely through basic algebraic expansion and exponent rules. This is the preferred method for dealing with trigonometry in electrical engineering and signal processing.