Six Trigonometric Functions (With Examples)

The six trigonometric functions are sine, cosine and tangent and their reciprocals: cosecant, secant and cotangent. In a right triangle, each one is a ratio of two sides for an acute angle $\theta$ . Cosecant flips sine, secant flips cosine, and cotangent flips tangent.

Before this lesson: You should know Trigonometric Ratios, the Pythagorean Theorem and Rationalizing the Denominator.

Three Ratios and Their Flips

In Geometry you used SOH CAH TOA for sine, cosine and tangent. Each one compares two sides of a right triangle, measured from an acute angle $\theta$ . Turning each fraction upside down gives three more ratios. Together they make the six trig functions.

The names follow a pattern. Each “co” function pairs with one that has no “co”: sine with cosecant, cosine with secant, and tangent with cotangent.

A right triangle with the right angle at the bottom right and angle theta at the bottom left. The side across from theta is labeled opposite, 5. The bottom side next to theta is labeled adjacent, 12. The longest side, across from the right angle, is labeled hypotenuse, 13. — For angle θ, the opposite side is 5, the adjacent side is 12 and the hypotenuse is 13.

Definitions of the Six Trig Functions

Function	Ratio	Reciprocal of
Sine	$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$	cosecant
Cosine	$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$	secant
Tangent	$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$	cotangent
Cosecant	$\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$	sine
Secant	$\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$	cosine
Cotangent	$\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$	tangent

\csc \theta = \frac{1}{\sin \theta} \qquad \sec \theta = \frac{1}{\cos \theta} \qquad \cot \theta = \frac{1}{\tan \theta}

Label the sides from the angle $\theta$ : opposite, adjacent and hypotenuse.
If a side is missing, find it with the Pythagorean theorem.
Write sine, cosine and tangent.
Flip each one to get cosecant, secant and cotangent.
Simplify each fraction and rationalize any square root in a denominator.

Worked Examples

Example 1: All six from a triangle

A right triangle has legs 5 and 12 and hypotenuse 13. The side opposite $\theta$ is 5. Find all six trig functions of $\theta$ .

Label the sides: opposite = 5, adjacent = 12, hypotenuse = 13

\sin \theta = \frac{5}{13}

Write sine.

\cos \theta = \frac{12}{13}

Write cosine.

\tan \theta = \frac{5}{12}

Write tangent.

\csc \theta = \frac{13}{5}

Flip sine.

\sec \theta = \frac{13}{12}

Flip cosine.

\cot \theta = \frac{12}{5}

Flip tangent.

Answer: $\sin \theta = \frac{5}{13}$ , $\cos \theta = \frac{12}{13}$ , $\tan \theta = \frac{5}{12}$ , $\csc \theta = \frac{13}{5}$ , $\sec \theta = \frac{13}{12}$ , $\cot \theta = \frac{12}{5}$

Example 2: Find the missing side first

In a right triangle, the side opposite $\theta$ is 8 and the side adjacent to $\theta$ is 15. Find all six trig functions of $\theta$ .

c^2 = 8^2 + 15^2

Use the Pythagorean theorem (see Pythagorean theorem)

c^2 = 64 + 225

Square each leg.

c^2 = 289

Add.

c = 17

Take the square root.

Write the three main ratios: $\sin \theta = \frac{8}{17}$ , $\cos \theta = \frac{15}{17}$ , $\tan \theta = \frac{8}{15}$

Flip them: $\csc \theta = \frac{17}{8}$ , $\sec \theta = \frac{17}{15}$ , $\cot \theta = \frac{15}{8}$

Answer: as listed in the last two steps.

Example 3: Start from one function

If $\theta$ is acute and $\sin \theta = \frac{3}{7}$ , find the other five trig functions.

Sine is opposite over hypotenuse, so draw a right triangle with opposite = 3 and hypotenuse = 7.

a^2 + 3^2 = 7^2

Set up the Pythagorean theorem for the adjacent side.

a^2 + 9 = 49

Square.

a^2 = 40

Subtract 9 from both sides.

a = \sqrt{40} = 2\sqrt{10}

Take the square root and simplify (see simplifying square roots)

\cos \theta = \frac{2\sqrt{10}}{7}

Write cosine.

\tan \theta = \frac{3}{2\sqrt{10}}

Write tangent.

\frac{3}{2\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{3\sqrt{10}}{20}

Rationalize the denominator (see rationalizing the denominator)

\csc \theta = \frac{7}{3}

Flip sine.

\sec \theta = \frac{7}{2\sqrt{10}} = \frac{7\sqrt{10}}{20}

Flip cosine and rationalize.

\cot \theta = \frac{2\sqrt{10}}{3}

Flip tangent.

Answer: $\cos \theta = \frac{2\sqrt{10}}{7}$ , $\tan \theta = \frac{3\sqrt{10}}{20}$ , $\csc \theta = \frac{7}{3}$ , $\sec \theta = \frac{7\sqrt{10}}{20}$ , $\cot \theta = \frac{2\sqrt{10}}{3}$

Example 4: Reciprocal functions on a calculator

Find $\sec 40^\circ$ and $\cot 25^\circ$ to the nearest hundredth.

\sec 40^\circ = \frac{1}{\cos 40^\circ}

Most calculators have no sec key, so use the reciprocal.

\cos 40^\circ \approx 0.76604

Evaluate cosine in degree mode.

\frac{1}{0.76604} \approx 1.3054

Divide.

\cot 25^\circ = \frac{1}{\tan 25^\circ}

Use the reciprocal for cotangent.

\tan 25^\circ \approx 0.46631

Evaluate tangent.

\frac{1}{0.46631} \approx 2.1445

Divide.

Answer: $\sec 40^\circ \approx 1.31$ and $\cot 25^\circ \approx 2.14$ . Calculator note: check that the calculator is in degree mode, then type 1 ÷ cos(40) and press ENTER. Do not use the cos⁻¹ key; that is the inverse function, not the reciprocal.

Example 5: Word problem with a ramp

A wheelchair ramp must rise 1 foot for every 12 feet of horizontal run. Let $\theta$ be the angle between the ramp and the ground. Find $\cot \theta$ and $\csc \theta$ , and explain what $\csc \theta$ tells a builder.

Label the sides: opposite = 1 (rise), adjacent = 12 (run)

c^2 = 1^2 + 12^2 = 1 + 144 = 145

Find the ramp length with the Pythagorean theorem.

c = \sqrt{145} \approx 12.04

Take the square root.

\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{12}{1} = 12

Write cotangent.

\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{\sqrt{145}}{1} \approx 12.04

Write cosecant.

Answer: $\cot \theta = 12$ and $\csc \theta = \sqrt{145} \approx 12.04$ . Cosecant is the length of ramp needed for each foot of rise, so a 2-foot rise needs about 24.08 feet of ramp.

Common Mistakes

Mistake	Example of the error	Fix
Pairing the wrong reciprocals	$\sec \theta = \frac{1}{\sin \theta}$	Each “co” pairs with a non-“co”: secant goes with cosine, cosecant with sine.
Using the inverse key for a reciprocal	Finding $\sec 40^\circ$ with cos⁻¹(40)	The reciprocal is $\frac{1}{\cos 40^\circ}$ . cos⁻¹ finds an angle, not a ratio.
Labeling sides from the wrong angle	Calling 12 the opposite side for $\theta$ in Example 1	Opposite and adjacent depend on which acute angle you use.
Leaving a radical in the denominator	$\tan \theta = \frac{3}{2\sqrt{10}}$	Rationalize: $\frac{3\sqrt{10}}{20}$ .
Calculator in radian mode	$\cos 40 \approx -0.667$	Switch to degree mode: $\cos 40^\circ \approx 0.766$ .

Related Methods

Check with the ranges

For an acute angle, sine and cosine are always less than 1, because the hypotenuse is the longest side. So cosecant and secant are always greater than 1. If you get $\sec \theta = 0.8$ , something went wrong.

Cofunctions of complementary angles

The two acute angles of a right triangle add to 90°. The side opposite one angle is adjacent to the other, so $\sin \theta = \cos (90^\circ - \theta)$ , $\tan \theta = \cot (90^\circ - \theta)$ and $\sec \theta = \csc (90^\circ - \theta)$ .

Any angle, not just acute angles

Later in this course, the six functions are defined for any angle using a point on the terminal side, so they work for angles like 150° or −60° too.

Practice Problems

A right triangle has opposite side 3, adjacent side 4 and hypotenuse 5 for angle $\theta$ . Find all six trig functions.
The side opposite $\theta$ is 7 and the side adjacent to $\theta$ is 24. Find $\csc \theta$ , $\sec \theta$ and $\cot \theta$ .
The hypotenuse is 10 and the side opposite $\theta$ is 6. Find all six trig functions.
If $\theta$ is acute and $\cos \theta = \frac{5}{13}$ , find the other five trig functions.
If $\theta$ is acute and $\tan \theta = 2$ , find the other five trig functions.
If $\csc \theta = 2$ , find $\sin \theta$ .
Find $\sec 70^\circ$ to the nearest hundredth.
Find $\cot 55^\circ$ to the nearest hundredth.
Find $\csc 12^\circ$ to the nearest hundredth.
A kite is 30 meters above the ground and 40 meters away horizontally from the person holding the string. Let $\theta$ be the angle between the string and the ground. Find $\sec \theta$ and explain what it means.

Show answers

Sine and cosine: $\sin \theta = \frac{3}{5}$ , $\cos \theta = \frac{4}{5}$

\tan \theta = \frac{3}{4}

Find the tangent: opposite over adjacent.

Flip each: $\csc \theta = \frac{5}{3}$ , $\sec \theta = \frac{5}{4}$ , $\cot \theta = \frac{4}{3}$
Answer: the six values above

c^2 = 7^2 + 24^2 = 49 + 576 = 625

Find the hypotenuse.

c = 25

Take the square root.

\csc \theta = \frac{25}{7}

Find the cosecant: hypotenuse over opposite.

\sec \theta = \frac{25}{24}

Find the secant: hypotenuse over adjacent.

\cot \theta = \frac{24}{7}

Find the cotangent: adjacent over opposite.

Answer: $\frac{25}{7}$ , $\frac{25}{24}$ , $\frac{24}{7}$

a^2 + 6^2 = 10^2

Find the adjacent side.

a^2 + 36 = 100

Square.

a^2 = 64

Subtract 36.

a = 8

Take the square root. A side length is positive.

Main ratios: $\sin \theta = \frac{6}{10} = \frac{3}{5}$ , $\cos \theta = \frac{8}{10} = \frac{4}{5}$ , $\tan \theta = \frac{6}{8} = \frac{3}{4}$
Flip: $\csc \theta = \frac{5}{3}$ , $\sec \theta = \frac{5}{4}$ , $\cot \theta = \frac{4}{3}$
Answer: the six values above

Draw a triangle with adjacent = 5 and hypotenuse = 13

b^2 + 5^2 = 13^2

Find the opposite side.

b^2 = 169 - 25 = 144

Subtract 25 from 169.

b = 12

Take the square root.

Main ratios: $\sin \theta = \frac{12}{13}$ , $\tan \theta = \frac{12}{5}$
Flip: $\csc \theta = \frac{13}{12}$ , $\sec \theta = \frac{13}{5}$ , $\cot \theta = \frac{5}{12}$
Answer: the five values above

Write tangent as a fraction: $\tan \theta = \frac{2}{1}$ , so opposite = 2 and adjacent = 1

c^2 = 2^2 + 1^2 = 5

Find the hypotenuse.

c = \sqrt{5}

Take the square root.

\sin \theta = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}

Find the sine and rationalize.

\cos \theta = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}

Find the cosine and rationalize.

Flip: $\csc \theta = \frac{\sqrt{5}}{2}$ , $\sec \theta = \sqrt{5}$ , $\cot \theta = \frac{1}{2}$
Answer: the five values above

\sin \theta = \frac{1}{\csc \theta}

Use the reciprocal.

\sin \theta = \frac{1}{2}

Substitute.

Answer: $\frac{1}{2}$

\sec 70^\circ = \frac{1}{\cos 70^\circ}

Use the reciprocal.

\cos 70^\circ \approx 0.34202

Evaluate.

\frac{1}{0.34202} \approx 2.9238

Divide.

Answer: about 2.92

\cot 55^\circ = \frac{1}{\tan 55^\circ}

Use the reciprocal.

\tan 55^\circ \approx 1.42815

Evaluate.

\frac{1}{1.42815} \approx 0.7002

Divide.

Answer: about 0.70

\csc 12^\circ = \frac{1}{\sin 12^\circ}

Use the reciprocal.

\sin 12^\circ \approx 0.20791

Evaluate.

\frac{1}{0.20791} \approx 4.8097

Divide.

Answer: about 4.81

10.

c^2 = 30^2 + 40^2 = 900 + 1600 = 2500

Find the string length.

c = 50

Take the square root.

\sec \theta = \frac{50}{40} = \frac{5}{4}

Secant is hypotenuse over adjacent.

Answer: $\sec \theta = \frac{5}{4}$ , so the string is 1.25 meters long for every meter of horizontal distance

FAQ

What are the six trigonometric functions?

They are sine, cosine, tangent, cosecant, secant and cotangent. The last three are the reciprocals of the first three.

How do you remember which functions are reciprocals?

Each pair has exactly one “co”: sine and cosecant, cosine and secant, tangent and cotangent.

Is csc the same as sin⁻¹?

No. $\csc \theta = \frac{1}{\sin \theta}$ is a ratio. The inverse sine, $\sin^{-1} x$ , gives the angle whose sine is $x$ .

How do you find secant on a calculator?

Find the cosine of the angle and divide 1 by it. For example, $\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{0.5} = 2$ .

Summary

Sine, cosine and tangent are opposite over hypotenuse, adjacent over hypotenuse and opposite over adjacent.
Cosecant, secant and cotangent are their reciprocals.
Each reciprocal pair has exactly one “co” in it.
Find a missing side with the Pythagorean theorem, then write all six ratios.
On a calculator, use 1 divided by sin, cos or tan.

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