Trigonometry is one of the essential topics in Mathematics from the exam point of view. It deals with finding various objects' angles, heights, and distances. The Greek Mathematician Hipparchus first coined the concept of trigonometry. Trigonometry includes trigonometric ratios trigonometrical identities that find out the measurements of the triangle's angles and find out the missing side of the triangle. 

This article focuses on Class 10 maths chapter 8 ‘Introduction to Trigonometry’ and its concepts in detail.

Trigonometrical ratios are the ratios of the sides of a triangle, particularly a right-angled triangle. The ratios measure the acute angles of the right-angled triangle. Listed below are the six types of trigonometric ratios:

• Sin θ, which is located on the opposite side to θ/hypotenuse.

• Cos θ, which is located on the adjacent side to the θ/hypotenuse.

• Tan θ, which is located on the opposite side/ adjacent side and the Sin θ and Cos θ.

• Sec θ, which is located on the opposite side to the hypotenuse/adjacent side and 1/cos θ.

• Cosec θ, which is located on the hypotenuse/opposite side and 1/sin θ.

• Cot θ, which is located on the adjacent side/opposite side and 1/tan θ.

Trigonometric ratios of some specific angles are the ratios of the right-angled triangles compared to the other acute angles. The angles of trigonometric ratios include 0°, 30°, 45°, 60°, and 90°. 

 

Suppose the triangle XYZ is an equilateral triangle. It is essential to draw a perpendicular line from X to determine the 30°and 60° angles. Now, after drawing the perpendicular line XD, the triangle is split into two triangles, XYD and DZX. 

 

Therefore, YD = DZ, and subsequently, angle YXD = angle DXZ according to the CPCT rule.

So triangle XYD ultimately becomes a right-angled triangle. According to the properties of the triangles where the sum of all internal angles should be 180°, angle YXD becomes 30°, and angle DXZ becomes 60°.

 

It is essential to know the measurements of the side lengths of the above triangles to find out the trigonometric ratio of the triangle. 

 

For example, XY = 2a. 

Then the sides YD = 1/2 and YZ = 1/2 = 2a = X since all the sides are identical in an equilateral triangle. 

 

Using Pythagoras Theorem, the sides XY² = YD²+XD² 

 

Therefore, XD² = (2a)² - a²

XD² = 4a² - a²

XD² = 3a²

Therefore, XY = a√3

So, the trigonometric ratios of 30° are: 

• Sin 30° = YD/XY = a/2a =½.

• Cos 30° = XD/XY = a√3 / 2a = √3/2

• Tan 30° = YD/XD = a/a√3 = 1/√3

• Cosec 30° = 1/sin 30° = 2

• Sec 30° = 1/cos 30° = 2/√3

• Cot 30° =  1/tan 30° = √3

Similarly, the trigonometric ratios of 60° are:

• Sin 60° = a√3/(2a) = √3/2

• Cos 60° = 1/2

• Tan 60° = √3

• Cosec 60° = 2/√3

• Sec 60° = 2

• Cot 60° =1/√3

Therefore, these are the trigonometric ratios of the angles 30° and 60°.

 

Suppose the triangle ABC is a right-angled triangle that has a right angle at the side B of the triangle. Since the angle on side B is 90°, the other angles must be 45° according to the properties of triangles where the sum of the interior angles of the triangle must be 180°. 

 

Thus, angles A and C are 45°. Subsequently, AB = BC, which becomes AB = BC =a. 

 

Using Pythagoras Theorem: 

AC² = AB² + BC² 

 

Therefore, AC² = a² + a² = 2a²

So the trigonometric ratios of the 45° angle are: 

• Sin 45°= BC/AC = a/ a√2 = 1/√2

• Cos 45°= AB/AC = a/ a√2 = 1/√2

• Tan 45°= BC/AB = a/a = 1

• Cosec 45°= 1/ sin 45° = √2

• Sec 45°= 1/ cos 45° = √2

• Cot 45°= 1/ tan 45° = 1

Consider that the triangle ABC mentioned above makes a right angle at side B. Angles A, B, and C are quite close to 0°. Therefore, Sin A = BC/AC is almost equal to 0° and Cos A = AB/AC = 1.

So, the trigonometrical ratios of 0° are:

• Sin 0° is 0 

• Cos 0°is 1 

• Tan 0° is 1/sin 0° and undefined

• Cosine 0° is 

• Sec 0° is  1/cos 0° = 1

• Cot 0° is 1/tan = 1/0° and undefined

 

Consider the same triangle ABC right angled at the side B. The numerical value of angle C is close to 0, and that of angle A is closer to 90°.

So, the trigonometrical ratios of 90° become: 

• Sin 90°= 1

• Cos 90°= 0

• Tan 90°= 90°/cos 90° = 1/0 = Undefined

• Sec 90°= 1/cos 90° = 1/0 = Undefined

• Cosec 90°= 1/sin 90° = 1

• Cot 90°= 1/tan 90° = 1/ ∞ = 0

Trigonometric identities are numerical values based on the trigonometrical ratios of right-angled triangles. The trigonometrical ratios of the triangles are sin, cos, cot, tan, and sec. 

 

Consider the same right-angled triangle ABC, with the right angle located in the side B. Therefore, the trigonometrical ratios of the triangle are:

 

• Sin C = AB/AC

• Cos C = BC/AC

• Cosec C = AC/AB

• Sec C = AC/BC

• Tan C = AB/BC

• Cot C = 1/tan C = BC/AB

The trigonometrical ratios of the angles 0°, 30°, 45°, 60°, 90° are as follows:

 

• For Sin:

• Sin C 0° = 0

• Sin C 30° = ½

• Sin C 45° = 1/√2

• Sin C 60° = √3/2

• Sin C 90° = 1

• For Cos:

• Cos C 0° = 1

• Cos C 30° = √3/2

• Cos C 45° = 1/√2

• Cos C 60° = 1/2

• Cos C 90° = 0

• For Tan:

• Tan C 0°= 0

• Tan C 30°= 1/√3

• Tan C 45°= 1

• Tan C 60°= √3

• Tan C 90°=  ∞

• For Cot:

• Cot C 0°=  ∞

• Cot C 30°= √3

• Cot C 45°= 1

• Cot C 60°= 1/√3

• Cot C 90°= 0

• For Sec:

• Sec C 0°= 0

• Sec C 30°= 2/√3

• Sec C 45°= √2

• Sec C 60°= 2

• Sec C 90°= ∞

• For Cosec:

• Cosec C 0°= ∞

• Cosec C 30°= 2

• Cosec C 45°= √2

• Cosec C 60°= 2/√3

• Cosec C 90°= 1

The trigonometric ratios of complementary angles refer to the sets of angles whose sum is 90°. For example, there are two angles, A and B, where angle A + angle B = 90°. 

 

Listed below are the trigonometric ratios of complementary angles:

• Sin (90° - θ) = Cos θ

• Cos (90° - θ) = Sin θ

• Tan (90° - θ) = Cot θ

• Cosec (90° - θ) = Sec θ

• Sec (90° - θ) = Cosec θ

• Cot (90° - θ) = Tan θ

Therefore, 

• Sin of the angle is the cos of its complementary angle.

• Tan of the angle is the cot of its complementary angle.

• Cos of the angle is the sin of its complementary angle.

• Cot of the angle is the tan of its complementary angle.

• Cosec of the angle is the sec of its complementary angle.

• Sec of the angle is the cosec of its complementary angle.

1. What is trigonometry and why is it important?

Trigonometry is the math concept that determines the values of angles using trigonometric ratios and identities. It is an important subject because it can be used in many real-life situations like plotting locations on a map, measuring celestial bodies and distances between objects, etc.

 

2. Name the angles in the trigonometric ratios.

The angles of the trigonometric ratios include sin, cos, cosec, sec, tan, and cot. These angles help determine the numerical value of the angles of a triangle through calculation. The trigonometric ratios of the angles also measure the unknown values of the triangles.
 

3. How can you find the angles in an equilateral triangle?

Firstly, draw the triangle and name all the sides. Draw a perpendicular line in between the triangle and split the triangle into two. This way, you can find the angles in an equilateral triangle.

So, for example, the name of the equilateral triangle is ABC. You can draw a perpendicular line AD from point A to line BC. This will split the triangle into two triangles: ABD and ADC.

      4. How can you find the angles in the right-angled triangle?

In a right-angled triangle, one of the angles is 90° due to the presence of a right angle in the triangle. Therefore, we can assume that the other two angles will be 45° due to the property that the sum of all the internal angles is 180°.

These are the trigonometric angles of the angles. There are several applications of trigonometry in real life like locating islands, making maps, measuring celestial bodies, and calculating heights and distances. Therefore, students must learn trigonometric ratios with the utmost concentration.