# Sides of a Triangle Formula

Before learning the sides of a triangle formula, let us recall a few things about a right-angled triangle. The sides of a right-angled triangle can be found out by using various methods like the Pythagoras theorem or by using the perimeter of the triangle. In case if some of the angles and other side lengths are given, we can use the law of cosines or the law of sines. In the following section, we will learn about the sides of a triangle formula.

## What Are the Sides of A Triangle Formula?

The sides of a triangle formula of a given triangle to find its sides are related to the trigonometric ratios. The necessary conditions include - one side of the triangle and an acute angle and thus, we can find out the rest of the sides of the triangle.

- In the case of a right triangle, we can apply the Pythagorean theorem directly.
- In the case of an isosceles triangle, we can use the area or perimeter formula.
- In case, some of the angles and other side lengths are known, we can use the law of cosines or the law of sines.

### Sides of A Triangle Formula

1. If we are given an angle and a side length for a right triangle,

- Sine θ = Length of the opposite side / Length of the Hypotenuse side
- Cos θ = Length of the adjacent side / Length of the Hypotenuse side
- Tan θ = Length of the opposite side / Length of the adjacent side

2. The law of sines: sin(A)/a = sin(B)/b = sin(C)/c

Where,

- a, b, and c are the sides of the triangles.
- A, B, and C are the corresponding angles.

3. The law of cosines: c^{2} = a^{2} + b^{2} - 2ab cos(C)

Where,

- a, b, and c are the sides of the triangles.
- C is the angle formed by the sides a and b.

Let us see the applications of sides of a triangle formula in the following section.

**Break down tough concepts through simple visuals.**

## Examples Using Sides of A Triangle Formula

**Example 1:** What are the sides of the right triangular park whose hypotenuse is 10 in and having a base angle of 30°?

**Solution:**

To find: The side of a triangular park.

Given,

Hypotenuse = 10 in.

The base angle = 30 degrees

Using sides of a triangle Formula,

Sine θ = Length of opposite side / Length of Hypotenuse side

Sine 30° = x/10

x = 5 in

And, Cos θ = Length of adjacent side / Length of Hypotenuse side

Cos 30°= y/10

y = 5√3 in

**Answer: The remaining sides of a triangular park are 5 in and 5√3 in.**

**Example 2: **What are the sides of the right triangle whose adjacent side is 10√3 in and having a base angle of 60°?

**Solution: **

To find: The length of one side of a triangular park.

Given,

Adjacent side = 10√3 in.

The base angle = 60 degrees

Using sides of a triangle Formula,

Cos θ = Length of adjacent side / Length of Hypotenuse side

Cos 60°= 10√3/y

y = 20√3 in

And, Sine θ = Length of opposite side / Length of Hypotenuse side

Sine 60° = x/20√3

x = 30 in

**Answer: The remaining sides of a triangular park are 30 in and 20√3 in.**

Example 3: Determine the sides of the triangle whose hypotenuse is 20 units and has a base angle of 60°?

**Solution:**

To find: The side of a triangle.

Given,

Hypotenuse = 20 units.

The base angle = 60 degrees

Using sides of a triangle Formula,

Sine θ = Length of opposite side / Length of Hypotenuse side

Sine 60° = x/20

√3/2 = x/20

x = 10√3 units

And, Cos θ = Length of adjacent side / Length of Hypotenuse side

Cos 60°= y/20

1/2 = y/20

y = 10 units

**Answer: The remaining sides of the triangle are 10√3 units and 10 units.**

## FAQs on Sides of A Triangle Formula

### What Is the Sides of a Triangle Formula?

The sides of a triangle formula to find the sides of a triangle using the trigonometric ratios. The sides of a triangle formulas are given as,

- Sine θ = Length of the opposite side / Length of the Hypotenuse side
- Cos θ = Length of the adjacent side / Length of the Hypotenuse side
- Tan θ = Length of the opposite side / Length of the adjacent side

In case if some of the angles and other side lengths of any triangle are given, we can use the law of cosines or the law of sines.

### When To Use Sides of a Triangle Formula?

We already that the sides of a triangle is applied only in the case of a right-angled triangle as it involves trigonometric ratios to be applied to find the sides of the given triangle. In case if some of the angles and other side lengths are given, we can use the law of cosines or the law of sines.

### How To Use Sides of a Triangle Formula?

If we are given an angle and a side length of a triangle,

- Step 1: Check for the type of triangle.
- Step 2: Check for any known sides or angles.
- Step 3: Put the given values in the sides of a triangle formula: Sine θ = Length of the opposite side / Length of the Hypotenuse side, Cos θ = Length of the adjacent side / Length of the Hypotenuse side, Tan θ = Length of the opposite side / Length of the adjacent side or directly use the law of sine or law of cosine accordingly.

### What Are the Applications of the Sides of a Triangle Formula?

The sides of a triangle formula has applications in trigonometry which further has a varied number of applications in our everyday life, from creation of maps to construction of buildings to what not,