# Sine vs. Cosine — What's the Difference?

By Fiza Rafique & Urooj Arif — Published on September 4, 2024
Sine and cosine are trigonometric functions that relate the angles of a triangle to the lengths of its sides in a right-angled triangle, sine represents the ratio of opposite side to the hypotenuse, and cosine the ratio of adjacent side to the hypotenuse.

## Key Differences

Sine (sin) and cosine (cos) are fundamental trigonometric functions that play a vital role in various areas of mathematics, physics, and engineering. The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. On the other hand, the cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Both functions are essential for calculating angles and distances in trigonometry and have a range of applications from signal processing to the analysis of periodic functions.
Sine and cosine functions are closely related and can be understood through the unit circle, where the value of sine is the y-coordinate of a point on the circle, while cosine is the x-coordinate. This relationship highlights how the two functions complement each other and are integral to understanding circular motion and oscillations.
In terms of their graphical representations, the sine function starts from 0 and goes up to 1 at 90 degrees, back to 0 at 180 degrees, down to -1 at 270 degrees, and returns to 0 at 360 degrees, forming a smooth, continuous wave. The cosine function, however, starts at 1 at 0 degrees, decreases to 0 at 90 degrees, goes to -1 at 180 degrees, returns to 0 at 270 degrees, and comes back to 1 at 360 degrees, also forming a wave but with a different phase from the sine wave.
The phase shift between the sine and cosine functions is precisely 90 degrees, meaning that sine can be thought of as a cosine wave shifted 90 degrees to the right, and vice versa. This property is used in analyzing waves and oscillations, where the phase difference can have physical significance.
Sine and cosine also have series expansions that allow them to be calculated for any angle, beyond just the acute angles in a right-angled triangle. These expansions reveal deeper mathematical properties and enable precise computations in various scientific and engineering disciplines.
The application of sine and cosine extends beyond triangles and circles; they are foundational in the study of periodic phenomena, such as sound and light waves, and in the analysis of harmonic motion in physics. Understanding the relationship and differences between sine and cosine is crucial for students and professionals in many technical fields.

## Comparison Chart

### Definition

Ratio of opposite side to hypotenuse in a right-angled triangle.
Ratio of adjacent side to hypotenuse in a right-angled triangle.

### Unit Circle Relation

Y-coordinate of a point on the unit circle.
X-coordinate of a point on the unit circle.

### Graphical Representation

Starts at 0, peaks at 1 and -1, and returns to 0 in a cycle.
Starts at 1, drops to -1, and returns to 1 in a cycle.

### Phase Shift

Can be considered as a cosine wave shifted 90 degrees left.
Can be considered as a sine wave shifted 90 degrees right.

### Series Expansion

Has an infinite series expansion for calculating its value.
Has an infinite series expansion for calculating its value.

## Compare with Definitions

#### Sine

Ratio of opposite side to hypotenuse in a triangle.
In a right-angled triangle, if the angle is 30 degrees, sin(30°) = 1/2.

#### Cosine

Ratio of adjacent side to hypotenuse in a triangle.
In a right-angled triangle, if the angle is 60 degrees, cos(60°) = 1/2.

#### Sine

Represents vertical displacement in circular motion.
The y-coordinate in unit circle representation is the sine of the angle.

#### Cosine

Represents horizontal displacement in circular motion.
The x-coordinate in unit circle representation is the cosine of the angle.

#### Sine

Infinite series expansion allows precise calculations.
Sin(x) can be computed for any x using its series expansion.

#### Cosine

Infinite series expansion for complex calculations.
Cos(x) can be computed for any angle x with its series expansion.

#### Sine

Essential for analyzing wave properties.
The height of a wave at any point can be determined using the sine function.

#### Cosine

Crucial for understanding rotational dynamics.
Cosine functions determine the position of rotating bodies.

#### Sine

Impacts oscillatory motion analysis.
Sine functions model the motion of pendulums and springs.

#### Cosine

Integral in wave phase analysis.
Cosine functions help analyze the phase difference between oscillating signals.

#### Sine

The ordinate of the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.

#### Cosine

In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.

#### Sine

In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.

#### Cosine

The abscissa at the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.

#### Sine

In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse. Category:en:Trigonometric functions Category:en:Functions

#### Cosine

(trigonometry) In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse. Symbol: cos Category:en:Functions Category:en:Trigonometric functions

#### Sine

The length of a perpendicular drawn from one extremity of an arc of a circle to the diameter drawn through the other extremity.

#### Cosine

Ratio of the adjacent side to the hypotenuse of a right-angled triangle

#### Sine

Ratio of the opposite side to the hypotenuse of a right-angled triangle

## Common Curiosities

#### How are sine and cosine used in real life?

They are used in fields such as engineering, physics, and architecture, for calculating distances, angles, and in analyzing waves and oscillations.

#### What does a phase shift mean for sine and cosine?

A phase shift refers to the horizontal movement of a wave form; sine and cosine functions are identical in shape but are shifted 90 degrees relative to each other.

#### What is the main difference between sine and cosine?

The main difference is their start points and their relationship to the sides of a triangle: sine relates to the opposite side/hypotenuse, while cosine relates to the adjacent side/hypotenuse.

#### Are sine and cosine only applicable to right-angled triangles?

While their definitions are based on right-angled triangles, their applications extend to any angle and to circular motion.

#### How do sine and cosine relate to the unit circle?

They correspond to the coordinates of a point on the unit circle: sine is the y-coordinate, and cosine is the x-coordinate.

#### Why are sine and cosine important in trigonometry?

They are fundamental in solving problems involving right-angled triangles and modeling periodic phenomena.

#### Can sine and cosine values be greater than 1 or less than -1?

No, the values of sine and cosine are always between -1 and 1, inclusive.

#### What happens to sine and cosine values as the angle approaches 360 degrees?

The functions are periodic, so as the angle approaches 360 degrees, the sine and cosine values return to their initial values at 0 degrees, completing a cycle.

#### What is the sine of 90 degrees and cosine of 0 degrees?

Sin(90°) = 1 and cos(0°) = 1, illustrating their maximum values.

#### How do you calculate sine and cosine without a calculator?

For specific angles (0°, 30°, 45°, 60°, 90°), values can be memorized. For other angles, their series expansions or trigonometric tables may be used.

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