Class 10 Maths Introduction to Trigonometry Notes

8.1 Introduction to Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of a triangle. The term Trigonometry comes from the Greek words “trigonon” (triangle) and “metron” (measure), meaning the study of the measures of triangles.

In particular, trigonometry is primarily concerned with right-angled triangles, where one of the angles is 90°. By using trigonometric ratios, we can relate the angles of a triangle to the lengths of its sides.

The three main elements of trigonometry are:

Trigonometric Ratios: These are ratios that relate the sides of a right-angled triangle to its angles.
Trigonometric Identities: These are mathematical expressions involving trigonometric functions that are true for all angles.
Trigonometric Values: These are specific values of trigonometric functions for known angles like 0°, 30°, 45°, 60°, and 90°.

Trigonometry is widely used in fields like engineering, physics, architecture, astronomy, and navigation.

8.2 Trigonometric Ratios

In a right-angled triangle, the trigonometric ratios are the ratios of the sides of the triangle relative to its angles. Given a right triangle with an angle $\theta$ θ, the sides are:

Hypotenuse (h): The side opposite the right angle, the longest side.
Opposite (o): The side opposite the angle $\theta$ θ.
Adjacent (a): The side next to the angle $\theta$ θ, but not the hypotenuse.

The six basic trigonometric ratios are:

Sine $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{o}{h}$ sinθ=HypotenuseOpposite=ho
Cosine $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{h}$ cosθ=HypotenuseAdjacent=ha
Tangent $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{o}{a}$ tanθ=AdjacentOpposite=ao
Cosecant $\csc \theta = \frac{1}{\sin \theta} = \frac{h}{o}$ cscθ=sinθ1=oh
Secant $\sec \theta = \frac{1}{\cos \theta} = \frac{h}{a}$ secθ=cosθ1=ah
Cotangent $\cot \theta = \frac{1}{\tan \theta} = \frac{a}{o}$ cotθ=tanθ1=oa

Example:

In a right triangle with angle $\theta = 30^\circ$ θ=30∘, the opposite side is 3 units and the adjacent side is 4 units. The hypotenuse is 5 units.

The trigonometric ratios will be:

$\sin 30^\circ = \frac{3}{5}$ sin30∘=53
$\cos 30^\circ = \frac{4}{5}$ cos30∘=54
$\tan 30^\circ = \frac{3}{4}$ tan30∘=43

These ratios are fundamental for solving trigonometric problems involving right-angled triangles.

8.3 Trigonometric Ratios of Some Specific Angles

In trigonometry, there are some commonly used angles for which the values of the trigonometric functions are fixed. These angles are $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ 0∘,30∘,45∘,60∘, and $90^\circ$ 90∘.

Here are the values of the six trigonometric functions for these angles:

Angle	$\sin \theta$ sinθ	$\cos \theta$ cosθ	$\tan \theta$ tanθ	$\csc \theta$ cscθ	$\sec \theta$ secθ	$\cot \theta$ cotθ
$0^\circ$ 0∘	0	1	0	Undefined	1	Undefined
$30^\circ$ 30∘	$\frac{1}{2}$ 21	$\frac{\sqrt{3}}{2}$ 23	$\frac{1}{\sqrt{3}}$ 31	2	$\frac{2}{\sqrt{3}}$ 32	$\sqrt{3}$ 3
$45^\circ$ 45∘	$\frac{\sqrt{2}}{2}$ 22	$\frac{\sqrt{2}}{2}$ 22	1	$\sqrt{2}$ 2	$\sqrt{2}$ 2	1
$60^\circ$ 60∘	$\frac{\sqrt{3}}{2}$ 23	$\frac{1}{2}$ 21	$\sqrt{3}$ 3	$\frac{2}{\sqrt{3}}$ 32	2	$\frac{1}{\sqrt{3}}$ 31
$90^\circ$ 90∘	1	0	Undefined	1	Undefined	0

These specific angle values are often used in problems involving standard angles and help simplify calculations.

8.4 Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the angle $\theta$ θ, where the functions involved are defined. Some important trigonometric identities are:

Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$ sin2θ+cos2θ=1 This identity is derived from the Pythagorean theorem applied to a right triangle.
Tangent and Secant Identity: $\tan^2 \theta + 1 = \sec^2 \theta$ tan2θ+1=sec2θ
Cotangent and Cosecant Identity: $1 + \cot^2 \theta = \csc^2 \theta$ 1+cot2θ=csc2θ

These identities are essential for simplifying and solving trigonometric equations. They are frequently used to prove other identities or solve trigonometric equations in both algebraic and geometric problems.

8.5 Summary

In this chapter, we learned:

Trigonometric Ratios: The six basic trigonometric ratios ( $\sin, \cos, \tan, \csc, \sec, \cot$ sin,cos,tan,csc,sec,cot) relate the sides of a right-angled triangle to its angles.
Trigonometric Ratios of Specific Angles: We studied the values of trigonometric functions for specific angles such as $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ 0∘,30∘,45∘,60∘, and $90^\circ$ 90∘.
Trigonometric Identities: We learned the fundamental identities like the Pythagorean identity and the relations between tangent, secant, cotangent, and cosecant.

Trigonometry is a crucial topic in mathematics, with applications ranging from geometry and physics to engineering and astronomy.

MCQs Based on the “Introduction to Trigonometry” Chapter:

1. Trigonometric Ratios:

Which of the following trigonometric ratios represents the ratio of the opposite side to the hypotenuse in a right-angled triangle?
a) $\cos \theta$ cosθ
b) $\sin \theta$ sinθ
c) $\tan \theta$ tanθ
d) $\cot \theta$ cotθ

Answer: b) $\sin \theta$ sinθ

2. Trigonometric Values of Specific Angles:

What is the value of tan⁡45∘\tan 45^\circtan45∘?
a) 0
b) 1
c) $\sqrt{3}$ 3
d) Undefined

Answer: b) 1

3. Pythagorean Identity:

Which of the following is a correct trigonometric identity?
a) $\sin^2 \theta + \cos^2 \theta = 0$ sin2θ+cos2θ=0
b) $\sin^2 \theta + \cos^2 \theta = 1$ sin2θ+cos2θ=1
c) $1 + \tan^2 \theta = \cos^2 \theta$ 1+tan2θ=cos2θ
d) $\cot^2 \theta = 1 + \sec^2 \theta$ cot2θ=1+sec2θ

Answer: b) $\sin^2 \theta + \cos^2 \theta = 1$ sin2θ+cos2θ=1

4. Trigonometric Ratios of Specific Angles:

For which of the following angles is sin⁡θ=1\sin \theta = 1sinθ=1?
a) $0^\circ$ 0∘
b) $30^\circ$ 30∘
c) $45^\circ$ 45∘
d) $90^\circ$ 90∘

Answer: d) $90^\circ$ 90∘

5. Trigonometric Ratios:

If tan⁡θ=13\tan \theta = \frac{1}{\sqrt{3}}tanθ=31, what is θ\thetaθ?
a) $30^\circ$ 30∘
b) $45^\circ$ 45∘
c) $60^\circ$ 60∘
d) $90^\circ$ 90∘

Answer: a) $30^\circ$ 30∘

6. Trigonometric Identities:

Which of the following is the correct form of the Pythagorean identity?
a) $\sin^2 \theta + \tan^2 \theta = 1$ sin2θ+tan2θ=1
b) $1 + \cot^2 \theta = \csc^2 \theta$ 1+cot2θ=csc2θ
c) $\sin^2 \theta + \cos^2 \theta = 2$ sin2θ+cos2θ=2
d) $\tan^2 \theta = 1 + \sec^2 \theta$ tan2θ=1+sec2θ

Answer: b) $1 + \cot^2 \theta = \csc^2 \theta$ 1+cot2θ=csc2θ

7. Trigonometric Functions:

Which of the following is the reciprocal of sin⁡θ\sin \thetasinθ?
a) $\cos \theta$ cosθ
b) $\sec \theta$ secθ
c) $\cot \theta$ cotθ
d) $\csc \theta$ cscθ

Answer: d) $\csc \theta$ cscθ

8. Trigonometric Ratios:

If the opposite side of a triangle is 4 units and the adjacent side is 3 units, what is tan⁡θ\tan \thetatanθ?
a) $\frac{4}{3}$ 34
b) $\frac{3}{4}$ 43
c) 1
d) $\frac{1}{2}$ 21

Answer: a) $\frac{4}{3}$ 34

9. Specific Angle Values:

The value of cos⁡60∘\cos 60^\circcos60∘ is:
a) 1
b) $\frac{1}{2}$ 21
c) $\frac{\sqrt{3}}{2}$ 23
d) 0

Answer: b) $\frac{1}{2}$ 21

10. Application of Trigonometric Identities:

Which of the following identities represents the relationship between tan⁡θ\tan \thetatanθ and sin⁡θ\sin \thetasinθ?
a) $\tan \theta = \frac{\sin \theta}{\cos \theta}$ tanθ=cosθsinθ
b) $\tan \theta = \frac{\cos \theta}{\sin \theta}$ tanθ=sinθcosθ
c) $\sin \theta = \frac{1}{\tan \theta}$ sinθ=tanθ1
d) $\tan \theta = \frac{1}{\sin \theta}$ tanθ=sinθ1

Answer: a) $\tan \theta = \frac{\sin \theta}{\cos \theta}$ tanθ=cosθsinθ