The Three Primary Trigonometric Comparisons

Introduction to the Three Primary Trigonometric Ratios

When ancient mathematicians studied triangles, they discovered useful patterns in the ratio of sides in right triangles. There are three primary trigonometric ratios that we will learn: sine ( $\sin$ ), cosine ( $\cos$ ), and tangent ( $\tan$ ).

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

Understanding the Sides of a Right Triangle

Before we go further, it's important to understand the terms used in trigonometric ratios:

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 side that is adjacent to the angle $\theta$ we are examining, not the hypotenuse.

Visualization of Triangle Sides

Move the slider to see how the position of sides changes with the angle.

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

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.

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

Examples of Sine Values

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

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.

Sin (60°) = 0.87Cos (60°) = 0.50Tan (60°) = 1.73

60°1.05 Radian

Examples of Cosine Values

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

Tangent

The tangent of angle $\theta$ is the ratio between the length of the opposite side and the length of the adjacent side. It can also be calculated as the ratio between the sine and cosine of the same angle.

\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}

Visualization of Tangent (

\tan \theta

)

Notice how the

\tan

value changes as the angle changes.

Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00

45°0.79 Radian

Examples of Tangent Values

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

Move the slider to see how the values of

\sin

\cos

, and

\tan

change on the unit circle.

Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00

45°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 $\sin \theta$ .
$\tan \theta$ is the slope of the line from the center to the point on the unit circle.

Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

\tan \theta = \frac{\sin \theta}{\cos \theta}

Exercise

Consider the following triangle with a $30^\circ$ angle:

Triangle with

30^\circ

Angle

Right triangle with a

30^\circ

angle

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

Check Your Answer

If the length of the hypotenuse is $1$ , then:

The value of $\sin 30^\circ$ is the opposite-side length, $0.5$
The value of $\cos 30^\circ$ is the adjacent-side length, $0.87$
The value of $\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{0.5}{0.87} = 0.58$

Introduction to the Three Primary Trigonometric Ratios

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

Understanding the Sides of a Right Triangle

Before we go further, it's important to understand the terms used in trigonometric ratios:

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 side that is adjacent to the angle $\theta$ we are examining, not the hypotenuse.

Visualization of Triangle Sides

Move the slider to see how the position of sides changes with the angle.

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

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.

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

Examples of Sine Values

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

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.

Sin (60°) = 0.87Cos (60°) = 0.50Tan (60°) = 1.73

60°1.05 Radian

Examples of Cosine Values

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

Tangent

\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}

Visualization of Tangent (

\tan \theta

)

Notice how the

\tan

value changes as the angle changes.

Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00

45°0.79 Radian

Examples of Tangent Values

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

Move the slider to see how the values of

\sin

\cos

, and

\tan

change on the unit circle.

Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00

45°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 $\sin \theta$ .
$\tan \theta$ is the slope of the line from the center to the point on the unit circle.

Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

\tan \theta = \frac{\sin \theta}{\cos \theta}

Exercise

Consider the following triangle with a $30^\circ$ angle:

Triangle with

30^\circ

Angle

Right triangle with a

30^\circ

angle

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

Check Your Answer

If the length of the hypotenuse is $1$ , then:

The value of $\sin 30^\circ$ is the opposite-side length, $0.5$
The value of $\cos 30^\circ$ is the adjacent-side length, $0.87$
The value of $\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{0.5}{0.87} = 0.58$