‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

Define the trigonometric ratios sine, cosine and tangent for an acute angle.

\(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\) (refer to the right-triangle figure in your NCERT textbook).

Write the values of \(\sin 30^\circ\), \(\cos 60^\circ\) and \(\tan 45^\circ\).

\(\sin 30^\circ = \tfrac{1}{2}\), \(\cos 60^\circ = \tfrac{1}{2}\) and \(\tan 45^\circ = 1\).

State the relationship between \(\tan\theta\), \(\sin\theta\) and \(\cos\theta\).

\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\).

Write the other two Pythagorean identities.

\(1 + \tan^2\theta = \sec^2\theta\) and \(1 + \cot^2\theta = \operatorname{cosec}^2\theta\).

Name the reciprocal ratios of sin, cos and tan.

\(\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}\), \(\sec\theta = \dfrac{1}{\cos\theta}\) and \(\cot\theta = \dfrac{1}{\tan\theta}\).

If \(\sin\theta = 3/5\), find \(\cos\theta\).

\(\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - 9/25} = \sqrt{16/25} = 4/5\).

Evaluate \(\sin 30^\circ \cos 60^\circ + \cos 30^\circ \sin 60^\circ\).

\(\left(\tfrac{1}{2}\right)\left(\tfrac{1}{2}\right) + \left(\tfrac{\sqrt{3}}{2}\right)\left(\tfrac{\sqrt{3}}{2}\right) = \tfrac{1}{4} + \tfrac{3}{4} = 1\).

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Chapter 8: Introduction to TrigonometryClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

Q: State the identity connecting \(\sin\theta\) and \(\cos\theta\).

\(\sin^2\theta + \cos^2\theta = 1\).

Q: State the fundamental trigonometric identity.

\(\sin^2\theta + \cos^2\theta = 1\).

Try one from this chapter

In a right triangle, sin θ is defined as:

Summary

Trigonometry studies the relationships between the angles and sides of right-angled triangles. The chapter defines the trigonometric ratios — sine, cosine, tangent and their reciprocals — for an acute angle, and gives their values for the standard angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ\) and \(90^\circ\). It also establishes fundamental trigonometric identities such as \(\sin^2\theta + \cos^2\theta = 1\), which are used to simplify expressions and prove results. These ratios and identities are the basis for solving a wide range of measurement problems.

Trigonometric ratiosRatios of standard anglesTrigonometric identities

Key terms

Trigonometric ratios

\(\sin,\ \cos,\ \tan\) and their reciprocals for an acute angle.

Identity \(\sin^2\theta + \cos^2\theta = 1\)

A fundamental relation true for all angles.

Standard angles

Angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) with known ratio values.

Reciprocal ratios

\(\operatorname{cosec},\ \sec\) and \(\cot\) — reciprocals of \(\sin,\ \cos\) and \(\tan\).

Extra questions & answers

EXERCISE 8.1

EXERCISE 8.2

EXERCISE 8.3

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sin, cos, tan on the unit circle

Angle θ45°

▮ sin▮ cos▮ radius (r = 1)

sin θ

0.707

cos θ

0.707

tan θ

sin, cos, tan on a right triangle

Angle θ30°

sin θ

0.5

cos θ

0.866

tan θ

0.5774

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\(\sin,\ \cos,\ \tan\) and their reciprocals for an acute angle.

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