The unit circle is fundamental to understanding trigonometric functions. For angle \(\theta\), we have:
\(\sin \theta = y\)-coordinate
\(\cos \theta = x\)-coordinate
\(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}\)
Key angles in radians and their values:
Interactive Unit Circle: Move the point around the circle to see how the coordinates change!
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: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
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:
\[\begin{align*} \tan(A + B) &= \frac{\tan A + \tan B}{1 - \tan A\tan B} \\ \tan(A - B) &= \frac{\tan A - \tan B}{1 + \tan A\tan B} \end{align*}\]Test your understanding of trigonometric concepts with interactive problems.