 # Trigonometric Ratios

The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). In geometry, trigonometry is a branch of mathematics that deals with the sides and angles of a right-angled triangle. Therefore, trig ratios are evaluated with respect to sides and angles.

The trigonometry ratios for a specific angle ‘θ’ is given below:

 Trigonometric Ratios Sin θ Opposite Side to θ/Hypotenuse Cos θ Adjacent Side to θ/Hypotenuse Tan θ Opposite Side/Adjacent Side & Sin θ/Cos θ Cot θ Adjacent Side/Opposite Side & 1/tan θ Sec θ Hypotenuse/Adjacent Side & 1/cos θ Cosec θ Hypotenuse/Opposite Side & 1/sin θ

Note: Opposite side is the perpendicular side and the adjacent side is the base of the right-triangle. Also, check out trigonometric functions to learn about each of these ratios or functions in detail.Trigonometric Identities

## Definition

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

The three sides of the right triangle are:

• Hypotenuse (the longest side)
• Perpendicular (opposite side to the angle)
• Base (Adjacent side to the angle)

## How to Find Trigonometric Ratios?

Consider a right-angled triangle, right-angled at B. With respect to ∠C, the ratios of trigonometry are given as:

• sine: Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.
• cosine: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.
• tangent: Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.
• cosecant: Cosecant is a multiplicative inverse of sine.
• secant: Secant is a multiplicative inverse of cosine.
• cotangent: Cotangent is the multiplicative inverse of the tangent.

The above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ ABC, the ratios are defined as:

sin C = (Side opposite to ∠C)/(Hypotenuse) = AB/AC

cos C = (Side adjacent to ∠C)/(Hypotenuse) = BC/AC

tan C = (Side opposite to ∠C)/(Side adjacent to ∠C) = AB/BC = sin ∠C/cos ∠C

cosec C= 1/sin C = (Hypotenuse)/ (Side Opposite to ∠C) AC/AB

sec C = 1/cos C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/BC

cot C = 1/tan C = (Side adjacent to ∠C)/(Side opposite to ∠C)= BC/AB

In right Δ ABC, if ∠and ∠C are assumed as 30° and 60°, then there can be infinite right triangles with those specifications but all the ratios written above for ∠C in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠A or ∠C) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle.

## Trigonometric Ratios Table

The trigonometric ratios for some specific angles such as 0 °, 30 °, 45 °, 60 ° and 90° are given below, which are commonly used in mathematical calculations.

 Angle 0° 30° 45° 60° 90° Sin C 0 1/2 1/√2 √3/2 1 Cos C 1 √3/2 1/√2 1/2 0 Tan C 0 1/√3 1 √3 ∞ Cot C ∞ √3 1 1/√3 0 Sec C 1 2/√3 √2 2 ∞ Cosec C ∞ 2 √2 2/√3 1

From this table, we can find the value for the trigonometric ratios for these angles. Examples are:

• Sin 30° = ½
• Cos 90° = 0
• Tan 45° = 1

### Trigonometry Applications

Trigonometry is one of the most important branches of mathematics. Some of the applications of trigonometry are:

• Measuring the heights of towers or big mountains
• Determining the distance of the shore from the sea
• Finding the distance between two celestial bodies
• Determining the power output of solar cell panels at different inclinations
• Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.

It is evident from the above examples that trigonometry has its involvement in a major part of our day-to-day life and much more. In most of the applications listed above, something was being measured and that is what trigonometry is all about.

## Solved Problems

Q.1: If in a right-angled triangle ABC, right-angled at B, hypotenuse AC = 5cm, base BC = 3cm and perpendicular AB = 4cm and if ∠ACB = θ, then find tan θ, sin θ and cos θ.

Sol: Given,

In ∆ABC,

Hypotenuse, AC = 5cm

Base, BC = 3cm

Perpendicular, AB = 4cm

Then,

tan θ = Perpendicular/Base = 4/3

Sin θ = Perpendicular/Hypotenuse = AB/AC = ⅘

Cos θ = Base/Hypotenuse = BC/AC = ⅗

Q.2: Find the value of tan θ if sin θ = 12/5 and cos θ = ⅗.

Sol: Given, sin θ = 12/5 and cos θ = ⅗

As we know,

Tan θ = Sin θ/Cos θ

Tan θ = (12/5)/(⅗)

Tan θ = 12/3

Tan θ = 4

## Practice Questions

1. Find the value of sin θ, if tan θ = ¾ and cos θ = ½.
2. Find tan θ if sin θ = 4/3 and cos θ = 3/2
3. Find sec θ, if cos θ = 9/8
4. Find cosec θ, if sin θ 16/5

## Video Lesson

### Trigonometric Ratios of Compound Angles Download BYJU’S App and learn thousands of concepts here through interesting and personalised videos.

## Frequently Asked Questions – FAQs

### What are the three primary trigonometric ratios?

The three primary trigonometric ratios are tangent (tan), sine (sin) and cosine (cos).

### What are the six trigonometric ratios?

The six 6 trigonometric ratios are sine, cosine, tangent, cotangent, cosecant, and secant.

### What is SOH CAH TOA?

SOH CAH TOA is the mnemonic to remember the formula for trigonometry ratios, such that:
SOH: Sine function is the ratio of Opposite side and Hypotenuse
CAH: Cosine is the ratio of Adjacent Side and Hypotenuse
TOA: Tangent is the ratio of Opposite side and Adjacent Side

### What is the formula for Cotangent, Secant and Cosecant?

Cotangent is the ratio of Adjacent side and Opposite side, (Base/Perpendicular)
Secant is the ratio of hypotenuse and adjacent side, (Hypotenuse/Base)
Cosecant is the ratio of hypotenuse and opposite side (Hypotenuse/Perpendicular)

### What is the relationship between sin, cos and tan?

Tangent functions is equal to the ratio of sine and cosine function.
Tan θ = Sin θ/Cos θ

Quiz on Trigonometric Ratios