Trigonometry Basics

What is trigonometry?

Trigonometry is the study of the relationship between the side lengths and angles of a triangle. It gets its name from the Greek trigonon (triangle) and metron (measure). In a right triangle, relative to an acute angle $\theta$ : the hypotenuse is the side opposite the right angle, the opposite side is opposite $\theta$ , and the adjacent side is next to $\theta$ .

The six ratios

The three main ratios are sine, cosine, and tangent, defined for an angle $\theta$ as:

\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}

Three more ratios are the reciprocals of these:

\csc\theta = \dfrac{1}{\sin\theta}, \quad \sec\theta = \dfrac{1}{\cos\theta}, \quad \cot\theta = \dfrac{1}{\tan\theta}

A key fact: the values of these ratios for a given angle don't depend on the size of the triangle, only on the angle itself, since any two right triangles with the same acute angle are similar (their sides are all in the same proportion). This is the foundation trigonometry is built on.

Worked example: finding a building's height

One common application is finding the height of a building without measuring it directly. Stand at a point $C$ , a distance of 250 m from the building ( $BC$ ), and measure the angle of elevation to the top of the building as 53°. Using the tangent function:

\tan 53^\circ = \dfrac{AB}{BC} = \dfrac{AB}{250}

Since $\tan 53^\circ \approx 1.327$ ,

1.327 = \dfrac{AB}{250} \implies AB = 331.75 \text{ m}

The building's height comes out to 331.75 m, without ever measuring it directly.

Standard angle values

It helps to know the ratios for a handful of standard angles: 0°, 30°, 45°, 60°, and 90°. There's a pattern worth noticing: for sine, the numerator (as $\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}$ over 2) increases from 0° to 90°; for cosine, the same five values repeat but in reverse. Tangent is just sine divided by cosine, and cosecant/secant/cotangent are the reciprocals of sine/cosine/tangent, so none of them need to be memorized separately.

	0°	30°	45°	60°	90°
sin θ	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
cos θ	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
tan θ	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	not defined
csc θ	not defined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$
sec θ	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	not defined
cot θ	not defined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$

Trigonometric identities

An identity is an equation involving trigonometric ratios that holds for every value of $\theta$ . The three Pythagorean identities:

$\sin^2\theta + \cos^2\theta = 1$
$\sec^2\theta - \tan^2\theta = 1$
$\csc^2\theta - \cot^2\theta = 1$

The double-angle and triple-angle identities:

$\sin 2\theta = 2\sin\theta\cos\theta$
$\cos 2\theta = \cos^2\theta - \sin^2\theta$
$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$
$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$
$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$
$\tan 3\theta = \dfrac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

Trigonometry Basics

What is trigonometry?

The six ratios

The three main ratios are sine, cosine, and tangent, defined for an angle $\theta$ as:

\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}

Three more ratios are the reciprocals of these:

\csc\theta = \dfrac{1}{\sin\theta}, \quad \sec\theta = \dfrac{1}{\cos\theta}, \quad \cot\theta = \dfrac{1}{\tan\theta}

Worked example: finding a building's height

\tan 53^\circ = \dfrac{AB}{BC} = \dfrac{AB}{250}

Since $\tan 53^\circ \approx 1.327$ ,

1.327 = \dfrac{AB}{250} \implies AB = 331.75 \text{ m}

The building's height comes out to 331.75 m, without ever measuring it directly.

Standard angle values

	0°	30°	45°	60°	90°
sin θ	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
cos θ	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
tan θ	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	not defined
csc θ	not defined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$
sec θ	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	not defined
cot θ	not defined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$

Trigonometric identities

An identity is an equation involving trigonometric ratios that holds for every value of $\theta$ . The three Pythagorean identities:

$\sin^2\theta + \cos^2\theta = 1$
$\sec^2\theta - \tan^2\theta = 1$
$\csc^2\theta - \cot^2\theta = 1$

The double-angle and triple-angle identities:

$\sin 2\theta = 2\sin\theta\cos\theta$
$\cos 2\theta = \cos^2\theta - \sin^2\theta$
$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$
$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$
$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$
$\tan 3\theta = \dfrac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

Trigonometry Basics

What is trigonometry?

The six ratios

Worked example: finding a building's height

Standard angle values

Trigonometric identities

Continue learning

Trigonometry Basics

What is trigonometry?

The six ratios

Worked example: finding a building's height

Standard angle values

Trigonometric identities

Continue learning