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.
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The opposite side is the side opposite to the angle a.
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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
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Formulas for Angle θ |
Reciprocal Identities |
Other Representations |
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sin θ = Opposite/Hypotenuse |
sinθ = 1/cosecθ |
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cos θ = Adjacent/Hypotenuse |
cosθ = 1/secθ |
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tan θ = Opposite/Adjacent |
tanθ = 1/cotθ |
tanθ= sinθ/ cosθ |
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cot θ = Adjacent/Opposite |
cotθ = 1/tanθ |
cotθ=cosθ/ sinθ |
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sec θ = Hypotenuse/Adjacent |
secθ = 1/cosθ |
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cosec θ = Hypotenuse/Opposite |
cosecθ = 1/sinθ |
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You can use the mnemonic SOH-CAH-TOA to remember the formulas easily.
The main values of the six trigonometric functions that we need to know are given below.
Values of all Trigonometric Functions
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Degrees |
Radians |
Sin θ |
Cos θ |
Tan θ |
Cot θ |
Sec θ |
Cosec θ |
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0⁰ |
0 |
0 |
1 |
0 |
Undefined |
1 |
Undefined |
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30⁰ |
Π/6 |
√3/2 |
1/√3 |
√3 |
2/√3 |
2 |
|
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45⁰ |
Π/4 |
1/√2 |
1/√2 |
1 |
1 |
√2 |
√2 |
|
60⁰ |
Π/3 |
√3/2 |
1/2 |
√3 |
1/√3 |
2 |
2/√3 |
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90⁰ |
Π/2 |
1 |
0 |
Undefined |
0 |
Undefined |
1 |
|
180⁰ |
Π |
0 |
-1 |
0 |
Undefined |
-1 |
Undefined |
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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?
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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.
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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.