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Sine, Cosine and Tangent

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What is Sine, Cosine and Tangent?

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Sine, cosine, and tangent (abbreviated as sin, cos, and tan) are three primary trigonometric functions, which relate an angle of a right-angled triangle to the ratios of two sides length. The reciprocals of sine, cosine, and tangent are the secant, the cosecant, and the cotangent respectively. Each of the six trigonometric functions has corresponding inverse functions (also known as inverse trigonometric functions). The trigonometric functions also known as the circular functions, angle functions, or goniometric functions are widely used in all fields of science that are related to Geometry such as navigation, celestial mechanics, solid mechanics, etc.

Read below to know what is a sine function, cosine function, and tangent function in detail.


Sine Cosine Tangent Definition

A right-angled triangle includes one angle of 90 degrees and two acute angles. Each acute angle of a right-angled triangle retains the property of the sine cosine tangent. The sine, cosine, and tangent of an acute angle of a right-angled triangle are defined as the ratio of two of three sides of the right-angled triangle.

As we know, sine, cosine, and tangent are based on the right-angled triangle, it would be beneficial to give names to each of the triangles to avoid confusion.


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  • “Hypotenuse side” is the longest side.

  • “Adjacent side” is the side next to angle θ.

  • “Opposite side” is the side opposite to angle θ.

Accordingly,

Sin θ = Opposite side/Hypotenuse

Cos θ = Adjacent/Hypotenuse

Tan θ = Opposite/Adjacent


What is the Sine Function?

In the right triangle, the sine function is defined as the ratio of the length of the opposite side to that of the hypotenuse side.

Sin θ = Opposite Side/ Hypotenuse Side.


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For example, the sine function of a triangle ABC with an angle θ  is expressed as:

Sin θ = a/c


What is the Cosine Function?

In the right triangle, the cosine function is defined as the ratio of the length of the adjacent side to that of the hypotenuse side.

Cos θ = Adjacent Side/Hypotenuse Side

Example:

Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:

Cos θ = b/c


What is the Tangent Function?

In the right triangle, the tangent function is defined as the ratio of the length of the opposite side to that of the adjacent side.

Tan θ = Opposite Side/Adjacent Side

Example:

Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:

Tan θ = a/b


Sine Cosine Tangent Table

The values of trigonometric ratios like sine, cosine, and tangent for some standard angles such as 0°, 30°, 45°, 60°, and 90° can be easily determined with the help of the sine cosine tangent table given below. These values are very important to solve trigonometric problems. Hence, it is important to learn the values of trigonometric ratios of these standard angles.


The sine, cosine, and tangent table given below includes the values of standard angles like 0°, 30°, 45°, 60°, and 90°.


Sine, Cosine, and Tangent Table

Angles In Degrees

30°

45°

60°

90°

Sin

0

1/2

1/√2

√3/2

1

Cos

1

√3/2

1/√2

1/2

0

Tan

0

1/√3

1

√3

Not Defined


Did You Know?

  • Sine and Cosine were introduced by Aryabhatta, whereas the tangent function was introduced by Muhammad Ibn Musa al- Khwarizmi ( 782 CE - 850 CE).

  • Sine Cosine and Tangent formulas can be easily learned using SOHCAHTOA. As sine is opposite side over hypotenuse side, cosine is adjacent side over hypotenuse side, and tangent is opposite side over the adjacent side.

Solved Examples:

1. Find Cos θ with respect to the following triangle.

Ans: To find Cos θ, we need both adjacent and hypotenuse side.


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The adjacent side in the above triangle is, BC = 8 Cm

But, the hypotenuse side i.e. AC  is not given.

To find the hypotenuse side, we use the Pythagoras theorem

AC² = AB² + BC²  =  6² + 8² = 100


Hypotenuse side, AC = √100 or 10 cm

Cos θ = Adjacent/Hypotenuse = 8/10

                               = 4/5

Therefore, Cos θ = 4/5


2. Find the value of  Sin 45°, Cos 60°, and Tan 60°.

Solution: Using the trigonometric table above, we have:

Sin 45°  = 1/√2

Cos 60° = 1/2

Tan 45°= 1

FAQ (Frequently Asked Questions)

Q1. How to Memorize Sine Cosine Tangent Values?

Ans. Steps to memorize sine, cosine tangent values:

  • The first step is to divide the numbers 0, 1, 2, 3, 4 by 4 and find a positive square root for the values obtained.

  • The angles in Sine Cosine Tangent are given in the order of 0°, 30°, 45°, 60°, and 90°.

  • You can remember the value of Sine-like this 0/√2, 1/√2, 2/√2, 3/√2, 4/√2.

  • The row of cosine is similar to the row of sine just in reverse order.

  • Each value of tangent can be obtained by dividing the sine values by cosine as Tan = Sin/Cos.

Q2. Why is the Sine Cosine Tan Function Important?

Ans. Sine cosine tan functions are important because of the following reason.

  • They help us to work on angles when sides of triangles are known.

  • They help us to work on sides of when angles of triangles are known.

Q3. What are the Applications of Trigonometry Function?

Ans. Some of the applications of trigonometric functions are:

  • Trigonometric functions are used in different fields like meteorology, seismology, physical Science, navigation, electronics, etc.

  • It is also used to measure the distance of long rivers, the height of mountains, etc.

  • The sine and cosine functions help to describe the sound and light waves.

  • It is used in satellite systems and also used for creating maps.