Trigonometric Ratios

Trigonometric Ratios: There are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratio between these sides based on the angle between them is called Trigonometric Ratio.

The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). Trigonometry, a branch of mathematics, focuses on the relationships between the sides and angles of right-angled triangles. Consequently, trigonometric ratios are determined based on these sides and angles.

As given in the figure in a right-angle triangle 

• The side opposite the right angle is called the hypotenuse
• The side opposite to an angle is called the opposite side
  • For angle C opposite side is AB
  • For angle A opposite side is BC
• The side adjacent to an angle is called the adjacent side
  • For angle C adjacent side is BC
  • For angle A adjacent side is AB

There are 6 basic trigonometric relations that form the basics of trigonometry. These 6 trigonometric relations are ratios of all the different possible combinations in a right-angled triangle.

These trigonometric ratios are called 

• Sine
• Cosine
• Tangent
• Cosecant
• Secant
• Cotangent

The mathematical symbol θ is used to denote the angle.

Sine of an angle is defined by the ratio of lengths of sides which is opposite to the angle and the hypotenuse. It is represented as sin θ

Cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. It is represented as cos θ

Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle. It is represented as tan θ

Cosecant of an angle is defined by the ratio of the length of the hypotenuse and the side opposite the angle. It is represented as cosec θ

Secant of an angle is defined by the ratio of the length of the hypotenuse and the side and the side adjacent to the angle. It is represented as sec θ

Cotangent of an angle is defined by the ratio of the length of sides that is adjacent to the angle and the side which is opposite to the angle. It is represented as cot θ.

Trigonometric ratios for any specific angle ‘θ’ is given below:

Trigonometric Ratios for standard angles 0°, 30°, 45°, 60°, and 90º are required to solve trigonometric ratios. Various trigonometric identities are also required to solve trigonometric functions. Now, the value of trigonometric ratios for specific angles such as 30°, 45°, etc in the table below:

This is one of the most basic and useful uses of trigonometry using the trigonometric ratios mentioned to find the length of a side of a right-angled triangle but to do, so we must already know the length of the other two sides or an angle and length of one side.

Steps to follow if one side and one angle are known:

1. Choose a trigonometric ratio that contains the given side and the unknown side
2. Use algebra to find the unknown side

Example: In a right-angled triangle, ABC ∠B = 90° and ∠C = 30° length of side AB is 4 find the length of BC given tan 30° = 1/√3. 

Solution:

C = 30°
tan C = tan 30°
         = 1/√3
tan C = opposite side/adjacent side
 1/√3 = AB/BC
 1/√3 = 4/BC
    BC = 4√3

Steps to follow if two sides are known:

1. Mark the known sides as adjacent, opposite, or hypotenuse with respect to any one of the acute angles in the triangle.
2. Decide on which trigonometric ratio can be found from the above table.
3. Find the angle (X)
4. Use a trigonometric ratio with respect to X which is a ratio of a known side and an unknown side.
5. Use algebra to find the unknown side.

Example: If two sides of a right-angled triangle are 20 and 10√3 where the side with length 20 is the hypotenuse, find the third side (without using Pythagoras theorem) given sin 30° = 1/2 and cos 30° = √3/2

Solution:

Given:

Hypotenuse =20, one side = 10√3, sin30° = 1/2, cos30° = √3/2

we can deduce that the angle opposite the side of length 10√3 is 30°, as sin 30° = 1/2 and cos30° = √3/2 are the values associated with 30°

using sine ratio:

sin(30°) = opposite/Hypotenuse

1/2 = opposite/20

Solving for opposite side, we get:

opposite = (1/2) × 20 = 10

So, length of third side of the triangle is 10 units.

Articles related to Trigonometric Ratios:

• Trigonometric Identites
• Real-life Application of trigonometry
• Sin and Cos Formulas

Example 1: Find the value of sin θ if tan θ = 4/3 and cos θ = 6/10.

Solution: 

Given,

tan θ = 4/3 and cos θ = 6/10

we know that,

tan θ = sin θ/cos θ

4/3 = sin θ/(6/10)

sin θ = (4/3)×(6/10)

sin θ = 8/10

Example 2: In a right-angled triangle PQR, right-angled at Q, the hypotenuse is PR = 13 cm, the base is QR = 5 cm and the perpendicular is PQ = 12 cm, and if ∠PQR = θ, then find sin θ, cos θ, and tan θ.

Solution: 

Given,

In ∆PQR,

Hypotenuse, PR = 13 cm

Base, QR = 5 cm

Perpendicular, PQ = 12 cm

now,

sin θ = Perpendicular/Hypotenuse = PQ/PR = 12/13

cos θ = Base/Hypotenuse = QR/PR = 5/13

tan θ = Perpendicular/Base = PQ/QR = 12/5

Example 3: Find the value of cos θ if tan θ = √3 and sin θ = √3/2.

Solution:

Given,

tan θ = √3 and sin θ = √3/2

we know that,

tan θ = sin θ/cos θ

√3 = (√3/2)/cos θ

cos θ = (√3/2)/√3

cos θ = 1/2

Example 4: Find the value of sec² θ if tan² θ = 1

Solution:

Given, tan² θ = 1…(1)

we know that, sec² θ – tan² θ = 1…(2)

By eq (1),

sec² θ – 1 = 1

sec² θ  = 1 + 1

than,

sec² θ  = 2

1. In a right-angled triangle, find the length of the side opposite to the angle θ if sinθ=0.6 and the hypotenuse is 10.

2. Given a right-angled triangle with cos⁡θ=0.8 and the adjacent side length is 6, find the hypotenuse.

3. Calculate tan⁡θ if the opposite side is 5 and the adjacent side is 12.

4. In a triangle with sec⁡θ=5/3 and the hypotenuse is 20, find the adjacent side.

5. Determine cot⁡θ if the adjacent side is 7 and the opposite side is 24.

Understanding trigonometric ratios and their applications is crucial for solving various problems involving right-angled triangles. These ratios provide a foundational toolset for more advanced topics in mathematics and its applications in science, engineering, and everyday problem-solving. Familiarity with the trigonometric table for standard angles further enhances the ability to quickly and accurately solve trigonometric problems.

What are various real-life applications of Trigonometric Ratios?

Various real life applications of trigonometric ratios are:

• sin and cos functions are used to represent various waves and harmonic functions.
• Trigonometric Ratios are used in the study of architectural engeering, civil engeering, astronomical stuies, etc.
• Trigonometric ratios are used in getting ideaof directions and creating maps.

Write the three main trigonometric ratios.

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

What are all the six trigonometric ratios?

All Six 6 trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).

Write the trigonometric ratios of complementary angles.

Two angles whose sum is 90° are called as complementary angles. The formulas for trigonometric ratios of complementary angles are:

• sin (90°- θ) = cos θ
• cos (90°- θ) = sin θ
• cosec (90°- θ) = sec θ
• sec (90°- θ) = cosec θ
• tan (90°- θ) = cot θ
• cot (90°- θ) = tan θ