Trigonometry

Unit Circle and Special Angles

The unit circle is fundamental to understanding trigonometric functions. For angle \(\theta\), we have:

\(\sin \theta = y\)-coordinate

\(\cos \theta = x\)-coordinate

\(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}\)

Special Angles and Values

Key angles in radians and their values:

\[\begin{array}{|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta \\ \hline 0 & 0 & 1 & 0 \\ \frac{\pi}{6} & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{3}} \\ \frac{\pi}{4} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 \\ \frac{\pi}{3} & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} \\ \frac{\pi}{2} & 1 & 0 & \text{undefined} \\ \hline \end{array}\]

Interactive Unit Circle: Move the point around the circle to see how the coordinates change!

Trigonometric Identities

Fundamental Identities

Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)

Reciprocal Identities:

\[\begin{align*} \csc \theta &= \frac{1}{\sin \theta} \\ \sec \theta &= \frac{1}{\cos \theta} \\ \cot \theta &= \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \end{align*}\]

Quotient Identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Double Angle Formulas

\[\begin{align*} \sin(2\theta) &= 2\sin \theta \cos \theta \\ \cos(2\theta) &= \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta \\ \tan(2\theta) &= \frac{2\tan \theta}{1 - \tan^2 \theta} \end{align*}\]

Half Angle Formulas

\[\begin{align*} \sin(\theta/2) &= \pm \sqrt{\frac{1 - \cos \theta}{2}} \\ \cos(\theta/2) &= \pm \sqrt{\frac{1 + \cos \theta}{2}} \\ \tan(\theta/2) &= \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} \end{align*}\]

Addition and Subtraction Formulas

Sine Addition and Subtraction:

\[\begin{align*} \sin(A + B) &= \sin A\cos B + \cos A\sin B \\ \sin(A - B) &= \sin A\cos B - \cos A\sin B \end{align*}\]

Cosine Addition and Subtraction:

\[\begin{align*} \cos(A + B) &= \cos A\cos B - \sin A\sin B \\ \cos(A - B) &= \cos A\cos B + \sin A\sin B \end{align*}\]

Tangent Addition and Subtraction:

\[\begin{align*} \tan(A + B) &= \frac{\tan A + \tan B}{1 - \tan A\tan B} \\ \tan(A - B) &= \frac{\tan A - \tan B}{1 + \tan A\tan B} \end{align*}\]

Ready to Practice?

Test your understanding of trigonometric concepts with interactive problems.