Key Points

In a right-angled triangle, for an acute angle $\theta$: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, and $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$. A common mnemonic is SOH-CAH-TOA.

The three reciprocal ratios are cosecant, secant, and cotangent. They are defined as $\operatorname{cosec} \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, and $\cot \theta = \frac{1}{\tan \theta}$.

The tangent and cotangent ratios can also be expressed as quotients of sine and cosine. The identities are $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

For any angle $\theta$, the most fundamental trigonometric identity is $\sin^2 \theta + \cos^2 \theta = 1$. This identity is true for all values of $\theta$.

Two other important identities derived from the main one are $1 + \tan^2 \theta = \sec^2 \theta$ and $1 + \cot^2 \theta = \operatorname{cosec}^2 \theta$. These are crucial for proving other trigonometric relations.

For a $45^\circ$ angle, the values are: $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\cos 45^\circ = \frac{1}{\sqrt{2}}$, and $\tan 45^\circ = 1$.

For a $30^\circ$ angle, the values are: $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

For a $60^\circ$ angle, the values are: $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, and $\tan 60^\circ = \sqrt{3}$.

For $0^\circ$: $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, $\tan 0^\circ = 0$. For $90^\circ$: $\sin 90^\circ = 1$, $\cos 90^\circ = 0$. Note that $\tan 90^\circ$ and $\sec 90^\circ$ are not defined.

If one trigonometric ratio is given, you can construct a right-angled triangle (e.g., if $\sin A = \frac{3}{5}$, opposite=3k, hypotenuse=5k). Use the Pythagorean theorem to find the third side, then calculate all other ratios.

For any acute angle $A$ ($0^\circ \leq A \leq 90^\circ$), the value of $\sin A$ and $\cos A$ is always between 0 and 1, inclusive. That is, $0 \leq \sin A \leq 1$ and $0 \leq \cos A \leq 1$.

For any acute angle $A$ ($0^\circ < A < 90^\circ$), the value of $\sec A$ and $\operatorname{cosec} A$ is always greater than or equal to 1. That is, $\sec A \geq 1$ and $\operatorname{cosec} A \geq 1$.

The notation $\sin A$ means 'the sine of angle A' and is not a product of 'sin' and 'A'. Similarly, $\sin^2 A$ is the standard way of writing $(\sin A)^2$.

The values of trigonometric ratios for an angle do not depend on the lengths of the sides of the triangle. They only depend on the measure of the angle, due to the properties of similar triangles.