The Three Primary Trigonometric Ratios

Side Ratios for a Fixed Acute Angle

For a fixed acute angle in a right triangle, corresponding side lengths always have the same ratios. The three primary ratios are sine ( $\sin$ ), cosine ( $\cos$ ), and tangent ( $\tan$ ).

\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}

Sides of a Right Triangle

Trigonometric ratios use the following terms:

Hypotenuse: The longest side of a right triangle, always opposite to the right angle $90^\circ$ .
Opposite side: The side that is opposite to the angle $\theta$ we are examining.
Adjacent side: The shorter side that touches the angle $\theta$ we are examining.

Visualization of Triangle Sides

Change the angle in the input below to see how the position of sides changes with the angle.

Sine

The sine of angle $\theta$ is the ratio between the length of the opposite side and the length of the hypotenuse.

\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}

Visualization of Sine (

\sin \theta

)

Notice how the

\sin

value changes as the angle changes.

Examples of Sine Values

The table lists the sine of the standard angles in exact form and as a decimal, so you can check a value in either representation.

Angle	Exact Sine Value	Approximate Decimal
$0^\circ$	$0$	$0$
$30^\circ$	$\frac{1}{2}$	$0.5$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$0.71$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$0.87$
$90^\circ$	$1$	$1$

Cosine

The cosine of angle $\theta$ is the ratio between the length of the adjacent side and the length of the hypotenuse.

\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}

Visualization of Cosine (

\cos \theta

)

Notice how the

\cos

value changes as the angle changes.

Examples of Cosine Values

The same standard angles appear here with their cosine values, and the decimal column gives the rounded value next to each exact one.

Angle	Exact Cosine Value	Approximate Decimal
$0^\circ$	$1$	$1$
$30^\circ$	$\frac{\sqrt{3}}{2}$	$0.87$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$0.71$
$60^\circ$	$\frac{1}{2}$	$0.5$
$90^\circ$	$0$	$0$

Tangent

The tangent of angle $\theta$ is the ratio between the opposite and adjacent sides. It is also the quotient of sine and cosine whenever $\cos \theta\ne0$ .

\begin{aligned} &\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}, \\ &\cos \theta\ne0 \end{aligned}

Visualization of Tangent (

\tan \theta

)

Notice how the

\tan

value changes as the angle changes.

Examples of Tangent Values

The table completes the three primary ratios with the tangent values for the same standard angles.

Angle	Exact Tangent Value	Approximate Decimal
$0^\circ$	$0$	$0$
$30^\circ$	$\frac{1}{\sqrt{3}}$	$0.58$
$45^\circ$	$1$	$1$
$60^\circ$	$\sqrt{3}$	$1.73$
$90^\circ$	$\text{undefined}$	$\text{undefined}$

Relationship between Sin Cos and Tan in the Unit Circle

To understand how these trigonometric ratios work for all angles, we can use the concept of the unit circle (a circle with radius $1$ ).

Unit Circle and Trigonometric Ratios

Change the angle in the input below to see how the values of

\sin

\cos

, and

\tan

change on the unit circle.

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

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

\tan\theta = 1

\theta = 45^\circ

0.79 Radian

In the unit circle:

The $x$ -coordinate on the unit circle is $\cos \theta$ .
The $y$ -coordinate on the unit circle is $\sin \theta$ .
When $\cos \theta\ne0$ , $\tan \theta$ is the slope of the line from the center to the point on the unit circle. At $90^\circ$ , that line is vertical, so its slope and tangent are undefined.

Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

\tan \theta = \frac{\sin \theta}{\cos \theta},\qquad \cos \theta\ne0

Exercise

The following right triangle has a $30^\circ$ angle and hypotenuse $1$ . Determine the exact opposite and adjacent side lengths, then find $\sin 30^\circ$ , $\cos 30^\circ$ , and $\tan 30^\circ$ .

Triangle with

30^\circ

Angle

Right triangle with a

30^\circ

angle

Worked Solution

The exact special-angle ratios give the two leg lengths directly:

\text{opposite side}=1\cdot\sin 30^\circ=\frac{1}{2}

Using exact values before rounding preserves the identity $\tan \theta=\sin \theta/\cos \theta$ and avoids error from dividing already rounded decimals.

Side Ratios for a Fixed Acute Angle

For a fixed acute angle in a right triangle, corresponding side lengths always have the same ratios. The three primary ratios are sine ( $\sin$ ), cosine ( $\cos$ ), and tangent ( $\tan$ ).

\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}

Sides of a Right Triangle

Trigonometric ratios use the following terms:

Hypotenuse: The longest side of a right triangle, always opposite to the right angle $90^\circ$ .
Opposite side: The side that is opposite to the angle $\theta$ we are examining.
Adjacent side: The shorter side that touches the angle $\theta$ we are examining.

Visualization of Triangle Sides

Change the angle in the input below to see how the position of sides changes with the angle.

Sine

The sine of angle $\theta$ is the ratio between the length of the opposite side and the length of the hypotenuse.

\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}

Visualization of Sine (

\sin \theta

)

Notice how the

\sin

value changes as the angle changes.

Examples of Sine Values

The table lists the sine of the standard angles in exact form and as a decimal, so you can check a value in either representation.

Angle	Exact Sine Value	Approximate Decimal
$0^\circ$	$0$	$0$
$30^\circ$	$\frac{1}{2}$	$0.5$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$0.71$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$0.87$
$90^\circ$	$1$	$1$

Cosine

The cosine of angle $\theta$ is the ratio between the length of the adjacent side and the length of the hypotenuse.

\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}

Visualization of Cosine (

\cos \theta

)

Notice how the

\cos

value changes as the angle changes.

Examples of Cosine Values

The same standard angles appear here with their cosine values, and the decimal column gives the rounded value next to each exact one.

Angle	Exact Cosine Value	Approximate Decimal
$0^\circ$	$1$	$1$
$30^\circ$	$\frac{\sqrt{3}}{2}$	$0.87$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$0.71$
$60^\circ$	$\frac{1}{2}$	$0.5$
$90^\circ$	$0$	$0$

Tangent

The tangent of angle $\theta$ is the ratio between the opposite and adjacent sides. It is also the quotient of sine and cosine whenever $\cos \theta\ne0$ .

\begin{aligned} &\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}, \\ &\cos \theta\ne0 \end{aligned}

Visualization of Tangent (

\tan \theta

)

Notice how the

\tan

value changes as the angle changes.

Examples of Tangent Values

The table completes the three primary ratios with the tangent values for the same standard angles.

Angle	Exact Tangent Value	Approximate Decimal
$0^\circ$	$0$	$0$
$30^\circ$	$\frac{1}{\sqrt{3}}$	$0.58$
$45^\circ$	$1$	$1$
$60^\circ$	$\sqrt{3}$	$1.73$
$90^\circ$	$\text{undefined}$	$\text{undefined}$

Relationship between Sin Cos and Tan in the Unit Circle

To understand how these trigonometric ratios work for all angles, we can use the concept of the unit circle (a circle with radius $1$ ).

Unit Circle and Trigonometric Ratios

Change the angle in the input below to see how the values of

\sin

\cos

, and

\tan

change on the unit circle.

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

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

\tan\theta = 1

\theta = 45^\circ

0.79 Radian

In the unit circle:

The $x$ -coordinate on the unit circle is $\cos \theta$ .
The $y$ -coordinate on the unit circle is $\sin \theta$ .
When $\cos \theta\ne0$ , $\tan \theta$ is the slope of the line from the center to the point on the unit circle. At $90^\circ$ , that line is vertical, so its slope and tangent are undefined.

Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

\tan \theta = \frac{\sin \theta}{\cos \theta},\qquad \cos \theta\ne0

Exercise

The following right triangle has a $30^\circ$ angle and hypotenuse $1$ . Determine the exact opposite and adjacent side lengths, then find $\sin 30^\circ$ , $\cos 30^\circ$ , and $\tan 30^\circ$ .

Triangle with

30^\circ

Angle

Right triangle with a

30^\circ

angle

Worked Solution

The exact special-angle ratios give the two leg lengths directly:

\text{opposite side}=1\cdot\sin 30^\circ=\frac{1}{2}

Using exact values before rounding preserves the identity $\tan \theta=\sin \theta/\cos \theta$ and avoids error from dividing already rounded decimals.