# Sine vs. Cosine: Know the Difference

By Shumaila Saeed || Updated on December 25, 2023

**Sine and cosine are trigonometric functions, sine representing the ratio of the opposite side to the hypotenuse in a right triangle, and cosine the ratio of the adjacent side to the hypotenuse.**

## Key Differences

Sine (sin) is a fundamental trigonometric function that measures the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. Cosine (cos), another key trigonometric function, measures the ratio of the length of the adjacent side to the angle to the hypotenuse.

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Nov 15, 2023

In the unit circle, sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle. Cosine, in contrast, is represented by the x-coordinate of this intersection point.

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The sine function is periodic, repeating every 2π radians or 360 degrees, and its values range between -1 and 1. Cosine function also shares these characteristics, but its waveform is phase-shifted by 90 degrees or π/2 radians.

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The graph of the sine function starts at the origin (0,0) and moves upwards, while the cosine graph starts at the maximum value of 1 when the angle is zero.

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In applications, sine is often used to model periodic phenomena such as sound waves. Cosine is similarly used, but its phase difference makes it suitable for different applications like alternating current waveforms.

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## Comparison Chart

### Definition in Right Triangle

Ratio of opposite side to hypotenuse.

Ratio of adjacent side to hypotenuse.

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### Unit Circle Representation

Y-coordinate of intersecting point.

X-coordinate of intersecting point.

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### Common Applications

Modeling vertical motion in waves.

Modeling horizontal motion in waves.

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## Sine and Cosine Definitions

#### Sine

Sine is the ratio of the opposite side to the hypotenuse in a right triangle.

The sine of 30 degrees is 0.5 in a right triangle.

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#### Cosine

In the unit circle, cosine is the x-coordinate of a point.

At 180 degrees, the cosine value is -1.

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#### Sine

In the unit circle, sine is the y-coordinate of a point.

At 90 degrees, the sine value reaches its maximum of 1.

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#### Cosine

Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle.

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#### Sine

A periodic function representing vertical oscillation in waveforms.

Sine waves are commonly used in acoustics to model sound.

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#### Cosine

A periodic function representing horizontal oscillation in waveforms.

Cosine waves are used to model the voltage in AC circuits.

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#### Sine

Sine function varies sinusoidally, repeating every 360 degrees.

The sine of an angle and its supplement are the same.

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#### Cosine

Cosine function has a sinusoidal variation, with a period of 360 degrees.

The cosine of an angle and its co-angle are the same.

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#### Sine

A function used in trigonometry to solve for unknown angles and sides.

Using sine, we can calculate the height of a tower from a known distance and angle.

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#### Cosine

A trigonometric function for calculating unknown angles and sides.

Cosine helps in determining the width of a river from a given point and angle.

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#### 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.

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#### Cosine

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

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#### Sine

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

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Oct 19, 2023

#### 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.

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Oct 19, 2023

#### 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

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Oct 19, 2023

#### 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

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#### Sine

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

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#### Cosine

The sine of the complement of an arc or angle. See Illust. of Functions.

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#### Cosine

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

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#### Sine

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

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## Repeatedly Asked Queries

#### Are sine and cosine values always between -1 and 1?

Yes, both functions have values ranging from -1 to 1.

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#### How are sine and cosine used in the unit circle?

Sine represents the y-coordinate and cosine the x-coordinate of a point on the unit circle.

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#### What is the period of sine and cosine functions?

Both functions have a period of 2π radians or 360 degrees.

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#### How are sine and cosine graphs different?

The sine graph starts from the origin, while the cosine graph starts from the maximum value.

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#### Are there any special angles for sine and cosine?

Yes, angles like 0°, 30°, 45°, 60°, and 90° have known sine and cosine values.

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#### What is the sine function?

Sine is a trigonometric function that relates an angle to the ratio of the opposite side to the hypotenuse in a right triangle.

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#### What is the cosine function?

Cosine is a trigonometric function that relates an angle to the ratio of the adjacent side to the hypotenuse in a right triangle.

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#### Can sine and cosine be used to solve triangles?

Yes, they are key in solving right triangles and in various applications of trigonometry.

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#### What is the phase shift between sine and cosine graphs?

There is a phase shift of 90 degrees or π/2 radians, with cosine leading.

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#### Can sine and cosine values be negative?

Yes, depending on the angle, both sine and cosine can take negative values.

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#### What are the maximum and minimum values of sine and cosine?

Both functions have maximum values of 1 and minimum values of -1.

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#### Are sine and cosine functions only for acute angles?

No, they can be used for any angle, including obtuse and reflex angles.

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#### Can sine and cosine be defined for complex numbers?

Yes, both functions can be extended to complex numbers.

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#### What is the significance of sine and cosine in Fourier series?

They are used to decompose periodic functions into their frequency components.

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#### How does the sine of an angle relate to its cosine?

The sine of an angle is the cosine of its complement, and vice versa.

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#### Are there inverse functions for sine and cosine?

Yes, the inverse sine (arcsin) and inverse cosine (arccos) functions exist.

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#### How do calculators compute sine and cosine?

Calculators use series approximations or algorithms to compute these values.

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#### How do sine and cosine relate to the exponential function?

They can be expressed using Euler's formula involving the exponential function.

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#### How are sine and cosine used in physics?

They are used to model waveforms, oscillations, and rotational motion.

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#### Is there a geometric interpretation of sine and cosine?

Yes, they can be interpreted geometrically using the unit circle.

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Nov 15, 2023

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About Author

Written by

Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.