# Trigonometry - Sine, Cosine and Tangent explained with worked examples

### Sine Cosine and Tangent:

Trigonometry is an aspect of mathematics that deals with triangles and the relationship between the 'three sides' and 'three angles' of triangles.

In Trigonometry, the right triangle is of interest to us because sine 'sin (θ)' and cosine 'cos (θ)' and tangent 'tan (θ)' are the three functions that reveal their shapes.

Before exploring these functions, let’s understand the names given to the three sides of a right triangle:

Note: The angle of the above right triangle is denoted as θ

Hypotenuse is always the longest side of the triangle.

Opposite is the side opposite the angle (θ)

Adjacent is the shorter side next to the angle (θ)

Each of Sin θ, cos θ and tan θ are the ratio of sides of the right triangle. They are calculated by using the formula below:

To get the ratio of the angle, make sure to always remember the magic word SOHCAHTOA

• SOH
• Sin = Opposite/Hypotenuse

• CAH

• TOA

Note: The functions sine, cosine and tangent helps us to easily calculate the sides of the triangle (, Opposite and Adjacent) when we know the angle.

Similarly, it also aid us to calculate the angle (sine, cosine and tangent) when we know the sides of the triangle.

Below are some worked Examples:

#### Example 1

Find the sine, cosine and tangent of 30° whose hypotenuse has a value of 2, opposite has a value of 1 and adjacent has a value of √3.

The image below illustrates the question.

Since we already know the length of each side of the triangle, the angles for sine, cosine and tangent can easily be calculated by implementing the keyword SOH-CAH-TOA

Sine = Opposite/Hypotenuse

Sin(30°) = 1 / 2 = 0.5

Cos(30°) = 1.732 / 2 = 0.866

Tangent

Tan(30°) = 1 / 1.732 = 0.577

Note: You may confirm the answers from a scientific calculator if you wish.

#### Example 2

Find “d” from the diagram below using the sine function.

Image Credit: Maths is Fun

From the figure above, we can conclude that:

• The angle is 39o

• The hypotenuse is 30m (Longest side of the triangle)

• d” is the opposite side of the triangle (side opposite the angle)

Applying the SOH-CAH-TOA formula, we will get:

Sin 39° = Opposite/Hypotenuse

Sin 39° = d/30

d = Sin 39° x 30

d = 0.6293 x 30

Note: Once we know the value of sine, cosine and tangent in a right triangle, we will be able to calculate the less common functions (Secant, Cosecant and Cotangent) using the formula below:

Secant Function:

sec(θ) = 1/cos

Cosecant Function:

csc(θ) = 1/sin

Cotangent Function:

cot(θ) = 1/tan

Alfred Ajibola is a Medical Biochemist, a passionate Academician with over 7 years of experience, a Versatile Writer, a Web Developer, a Cisco Certified Network Associate and a Cisco CyberOps Associate.

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