Trigonometric Functions

Circular Functions and Trigonometry 

Trigonometry is the study of the relationships between the sides and angles of a right-angled triangle. This concept was first introduced by a Greek mathematician, Hipparchus.

What is a Trigonometric Function?

Circular functions (also called trigonometric functions or trigonometric ratios) are the functions or ratios which state the relationship between the angles and sides of a right-angled triangle.

Basic Trigonometric Functions (sin, cos, tan, cosec, sec, cot)

Sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot) are the basic trigonometric functions. Of these, sine, cosec, and tangent are the primary trigonometric functions.

To define the trigonometric functions, we use the diagram of a right-angled triangle, that is called the sin-cos-tan diagram in trigonometry.

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  • The hypotenuse is the longest side of a right-angled triangle, and it is always opposite to the right angle.

  • The opposite side is the side opposite to the angle a.

  • The adjacent side is the side joining angle a with the right angle.

Definition of the Six Trigonometric Functions

Keeping this diagram in mind, we can now define the primary trigonometric functions.

1. Sine (sin): Sine function of an angle (theta) is the ratio of the opposite side to the hypotenuse.  

2. Cosine (cos): Cosine function of an angle (theta) is the ratio of the adjacent side to the hypotenuse.  

3. Tangent (tan): The tangent function of an angle (theta) is the ratio of the opposite side to the adjacent side.  

The following functions are derived from and are the reciprocals of sine, cosine, and tangent.

4.Cosecant (cosec): Cosecant is the reciprocal of sine. Cosecθ is basically the hypotenuse divided by the opposite side.

5.Secant (sec): Secant is the reciprocal of cosine. Secθ is basically the hypotenuse divided by the adjacent side.

6.Cotangent (cot): Cotangent is the reciprocal of the tangent. Cotθ is basically the adjacent side divided by the opposite side.  

Six Trigonometric Functions Chart

Formulas for Angle θ

Reciprocal Identities

Other Representations

sin θ = Opposite/Hypotenuse

sinθ = 1/cosecθ

 

cos θ = Adjacent/Hypotenuse

cosθ = 1/secθ

 

tan θ = Opposite/Adjacent

tanθ = 1/cotθ

tanθ= sinθ/ cosθ

cot θ = Adjacent/Opposite

cotθ = 1/tanθ

cotθ=cosθ/ sinθ

sec θ = Hypotenuse/Adjacent

secθ = 1/cosθ

 

cosec θ = Hypotenuse/Opposite

cosecθ = 1/sinθ

 

 

You can use the mnemonic SOH-CAH-TOA to remember the formulas easily.

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The main values of the six trigonometric functions that we need to know are given below.

Values of all Trigonometric Functions

Degrees

Radians

Sin θ

Cos θ

Tan θ

Cot θ

Sec θ

Cosec θ

0⁰

0

0

1

0

Undefined

1

Undefined

30⁰

Π/6


√3/2

1/√3

√3

2/√3

2

45⁰

Π/4

1/√2

1/√2

1

1

√2

√2

60⁰

Π/3

√3/2

1/2

√3

1/√3

2

2/√3

90⁰

Π/2

1

0

Undefined

0

Undefined

1

180⁰

Π

0

-1

0

Undefined

-1

Undefined

360⁰

2/Π

0

1

0

Undefined

1

Undefined

Trigonometric Functions Examples With Solution

Example 1: A bird is sitting on a tree at an angle of elevation of 20°. A boy is standing 10 metres away from the tree. What is the distance between the bird and the ground?

Solution

Angle of Elevation: θ = 20°

Distance between boy and tree: Opposite Side=10m

Distance between bird and ground: Adjacent Side= x

The tangent function gives the relationship between θ, opposite side and adjacent side.

tan θ = opposite side/adjacent side

tan (20°) = 10/x

x=10/tan(20°)

x = 10 × 0.36

x= 3.6 m

Therefore, the bird is sitting at a height of 3.6 m from the ground.

Example 2: A fruit is on a tree at a height of 3 m. A girl is standing 3 metres away from the tree. The girl puts a ladder from where she is standing to the fruit. What is the angle between the ladder and the tree? 

Solution:

The angle between the ladder and the tree= x

Adjacent Side= 3 m

Opposite Side= 3 m

tan θ = opposite side/adjacent side

tan (x)= 3/3

tan(x)= 1

x= 45° 

Therefore, the angle between the ladder and the tree is 45°.

Did you know? 

  • Trigonometry is used by marine biologists to determine the amount of sunlight that influences photosynthesis in algae. Using trigonometric ratios and statistical frameworks, marine biologists can predict the scale and behaviour of bigger species such as whales.

  • Trigonometry is also used to investigate a crime scene. The fundamentals of trigonometry are valuable in determining the course of projectile motion and in predicting the effects of a crash in a vehicle accident. It is often used to describe how a particular target fell or at what position the bullet was fired.

FAQ (Frequently Asked Questions)

1. What are Some Applications of Trigonometry?

Most of the design and engineering depends on triangular support. When an engineer measures the length of the cables, the height of the support poles, and the angle between the two while calculating weight loads and bridge power, trigonometry allows him to measure the right angles. It also enables the builders to correctly lay out a curved wall, or to figure out the appropriate slope for the roof. Thus, trigonometry plays a major role in automotive engineering, enabling car companies to size every part precisely and ensure that the machine components work safely together.

 2. How is Trigonometry Related to Waves?

When they are graphed, the trigonometric functions follow a basic wave pattern. Sound waves, light waves, even ocean waves and electrical currents can be represented using the trigonometric ratios. Thus, trigonometry plays a key role in musical theory and production. The visual depiction of music helps computers in recognising sounds. It also helps sound engineers to imagine sound waves in such a way that they can change volume, pitch and other variables to produce the desired sound effects. Trigonometry also plays a central role in the placement of speakers, as the angles of the sound waves when they reach the ears can influence the sound quality.