Trigonometry

SOH CAH TOA: Sine, Cosine and Tangent in Right-Angled Triangles

In a right-angled triangle, sine, cosine and tangent are ratios of sides that depend only on an angle: $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$. SOH CAH TOA is the memory aid.

What Is a Radian? Converting Between Degrees and Radians

A radian is the angle at the centre of a circle where the arc length equals the radius. A full turn is $2\pi$ radians, so $180° = \pi$ radians.

The Unit Circle Explained: Sine and Cosine for Any Angle

The unit circle has radius $1$ and centre at the origin. The point at angle $\theta$ around it has coordinates $(\cos\theta, \sin\theta)$, which defines sine and cosine for every angle, not just those under $90°$.

Graphs of Sine and Cosine: Amplitude, Period and Shifts

The graphs of $y = \sin x$ and $y = \cos x$ are waves that repeat every $2\pi$. In $y = A\sin(B(x - h)) + k$, $|A|$ is the amplitude, $\frac{2\pi}{|B|}$ is the period, and $h$ and $k$ shift the wave.

Trig Identities: sin²θ + cos²θ = 1 and the Double-Angle Formulas

A trig identity is an equation that is true for every angle. The most important, $\sin^2\theta + \cos^2\theta = 1$, is Pythagoras on the unit circle; others include $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and the double-angle formulas.

Solving Trig Equations and Inverse Trig Functions

To solve a trig equation like $\sin x = \frac12$, use an inverse trig function to find one angle, then use the symmetry of the unit circle to find every other angle in the range you need.