INTRODUCTION TO TRIGONOMETRY-True/False

Trigonometry in Class 10 begins with studying the ratios of sides of a right triangle with respect to its acute angles. Students learn six basic trigonometric ratios: sine, cosine, tangent and their reciprocals cosecant, secant and cotangent, all defined using perpendicular, base and hypotenuse of a right-angled triangle. The chapter also develops fundamental identities such as \({\mathbf{sin}^\mathbf{2}\theta}+{\mathbf{cos}^\mathbf{2}\theta}=\mathbf{1}, {\mathbf{sec}^\mathbf{2}\theta}-{\mathbf{tan}^\mathbf{2}\theta}=\mathbf{1} \text{ and } {\mathbf{\text{cosec }}^\mathbf{2}\theta}-{\mathbf{cot}^\mathbf{2}\theta}=\mathbf{1}\), which are frequently used in numerical problems and proofs. Another important focus is on trigonometric ratios of standard angles \(\mathbf{0}^\circ,{\mathbf{30}}^\circ,{\mathbf{45}}^\circ,{\mathbf{60}}^\circ \text{ and } {\mathbf{90}}^\circ,\) which students must memorise accurately for quick problem-solving in board examinations. The chapter also explores complementary angle relationships like \(\mathbf{sin}({\mathbf{90}}^\circ-\ {\theta})=\mathbf{cos}\ {\theta}, \mathbf{tan}({\mathbf{90}}^\circ-\ {\theta})=\mathbf{cot}\ {\theta} \) and similar results for the remaining ratios. Carefully designed True/False questions on these concepts help learners avoid common mistakes, strengthen formula recall, and develop confidence to tackle higher-level trigonometry in later classes.

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TRIGONOMETRIC FUNCTIONS-Exercise 3.2

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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TRIGONOMETRIC FUNCTIONS-Exercise 3.1

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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INTRODUCTION TO TRIGONOMETRY

by Academia Aeternum

1. In a right triangle, if an acute angle is \(\theta\), then \(\sin \theta = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}}\).
2. In a right triangle, \(\cos \theta = \dfrac{\text{Perpendicular}}{\text{Base}}\).
3. For an acute angle \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\).
4. The reciprocal of \(\sin \theta\) is \(\sec \theta\).
5. For any acute angle \(\theta?\) in a right triangle, \(\sin^2 \theta + \cos^2 \theta = 1\).
6. For any acute angle \(\theta\ \sec^2 \theta - \tan^2 \theta = 1\).
7. For any acute angle \(\theta\), \(\text{cosec}^2 \theta - \cot^2 \theta = 1\).
8. \(\tan \theta \cdot \cot \theta = 1\) for any acute angle \(\theta\).
9. \(\sin 0^\circ = 1\) and \(\cos 0^\circ = 0\).
10. \(\sin 90^\circ = 1\) and \(\cos 90^\circ = 0\).
11. \(\sin 30^\circ = \dfrac{1}{2}\) and \(\cos 30^\circ = \dfrac{\sqrt{3}}{2}\).
12. \(\sin 45^\circ = \dfrac{1}{\sqrt{2}}\) and \(\cos 45^\circ = \dfrac{1}{\sqrt{2}}\).
13. \(\tan 45^\circ = 1\)
14. \(\tan 30^\circ = \sqrt{3}\) and\(\tan 60^\circ = \dfrac{1}{\sqrt{3}}\).
15. \(\sin 60^\circ = \dfrac{\sqrt{3}}{2}\) and \(\cos 60^\circ = \dfrac{1}{2}\).
16. For any angle \(\theta\) (where defined), \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \dfrac{\cos \theta}{\sin \theta}\).
17. For complementary angles, \(\sin (90^\circ - \theta) = \cos \theta\).
18. For complementary angles, \(\tan (90^\circ - \theta) = \tan \theta\).
19. For complementary angles, \(\\text{cosec } (90^\circ - \theta) = \sec \theta\) and \(\cot (90^\circ - \theta) = \tan \theta\).
20. In a right triangle, the hypotenuse is always the longest side.
21. In a right triangle, for an acute angle \(\theta\), \(\sin \theta\) can be greater than 1.
22. If sin?\(\sin \theta = \dfrac{3}{5}\) for an acute angle ?\theta?, then cos?\(\cos \theta = \dfrac{4}{5}\).
23. If \(\cos \theta = \dfrac{5}{13}\) for an acute angle \(\theta\), then \(\sin \theta = \dfrac{12}{13}\).
24. The trigonometric ratios introduced in Class 10 Chapter 8 are defined only for acute angles of a right triangle.
25. In Class 10, the standard angles for which exact trig values are usually memorised are \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\).

Frequently Asked Questions

Trigonometry is the branch of mathematics that studies the relationship between the sides and angles of a right-angled triangle usin g trigonometric ratios such as sin e, cos in e, and tan gent.

Trigonometric ratios are ratios of the lengths of the sides of a right triangle with respect to one of its acute angles. They include sin , cos , tan , cos ec, sec , and cot .

The six ratios are: sin \(\theta\), cos \(\theta\), tan \(\theta\), cos ec\(\ \theta\), sec \(\theta\), and cot \(\theta\).

sin \(\theta\) = Opposite side ÷ Hypotenuse.

cos \(\theta\) = Adjacent side ÷ Hypotenuse.

tan \(\theta\) = Opposite side ÷ Adjacent side.

tan \(\theta\) = sin \(\theta\) ÷ cos \(\theta\).

cosec\(\ \theta\) = 1 ÷ sin \(\theta\) = Hypotenuse ÷ Opposite side.

sec \(\theta\) = 1 ÷ cos \(\theta\) = Hypotenuse ÷ Adjacent side.

cot \(\theta\) = 1 ÷ tan \(\theta\) = Adjacent side ÷ Opposite side.

Values include: sin 0\(^\circ\)=0, sin 30\(^\circ\)=1/2, sin 45\(^\circ\)=v2/2, sin 60\(^\circ\)=v3/2, sin 90\(^\circ\)=1 (others similarly defined).

They help solve real-life problems involving heights, distan ces, angles of elevation/depression, navigation, physics, engineering, and architecture.

The angle formed between the horizontal line and the line of sight when the observer looks upward at an object.

The angle formed between the horizontal line and the line of sight when the observer looks downward from a higher point.

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

1 + tan ²\(\ \theta\) = sec ²\(\ \theta\) and 1 + cot ²\(\ \theta\) = cos ec²\(\ \theta\).

tan \(\theta\) × cot \(\theta\) = 1.

sin \(\theta\) × cosec\(\ \theta\) = 1.

cos \(\theta\) × sec \(\theta\) = 1.

45\(^\circ\), because sin 45\(^\circ\) = cos 45\(^\circ\) = v2/2.

Only acute angles (0\(^\circ\) < \(\theta\) < 90\(^\circ\)) are considered in this chapter.

No, negative angles and circular trigonometric functions are not introduced at this level.

Trigonometric ratios help determine unknown heights, widths, and distan ces by relating them to measured angles.

tan \(\theta\) = sin \(\theta\)/cos \(\theta\).

Use the mnemonic: SOH-CAH-TOA.or Pundit Badri Prasad Har Har Bole

Because it lies opposite the right angle, based on the Pythagorean theorem.

tan 90\(^\circ\) is undefined because cos 90\(^\circ\)=0.

sec 90\(^\circ\) = 1/cos 90\(^\circ\) = undefined.

Because tan 0\(^\circ\)=0, so cot 0\(^\circ\) = 1/0, which is undefined.

Two angles whose sum is 90\(^\circ\).

sin (90\(^\circ\)-\(\ \theta\))=cos \(\theta\), cos (90\(^\circ\)-\(\ \theta\))=sin \(\theta\), tan (90\(^\circ\)-\(\ \theta\))=cot \(\theta\), etc.

Ratios depend only on the angle, not on the actual size of the triangle.

No. For a given angle, the trigonometric ratios remain constan t.

\(\sin \theta\) = tan \(\theta\sqrt{(1+tan ²\ \theta}\).

\(\cos \theta = \dfrac{1}{\sqrt{(1+tan ² \theta}}\).

\(\tan \theta = \dfrac{\sin \theta}{\sqrt{(1-sin^2\ \theta}}\).

\(\tan \theta= \dfrac{\sqrt{(1-cos^2\ \theta)}}{\cos \theta}\).

Problems involving complementary angles and stan dard value tables are frequently tested.

Measuring mountain heights, building construction, aviation, satellite tracking, GPS, and navigation.

Class 10 NCERT curriculum covers only degree measure for introductory learning.

Yes, but such applications require advanced formulas (sin e rule, cos in e rule) taught in higher classes.

Theodolites, sextan ts, clinometers, laser rangefinders, and surveying instruments.

A surveying instrument used to measure horizontal and vertical angles for calculating heights and distan ces.

It is used in designing structures, mechanical components, electrical waves, circuits, and digital systems.

It creates equal opposite and adjacent sides, leading to simple trigonometric values.

Three core identities: sin ²\(\ \theta\) + cos ²\(\ \theta\) = 1; 1 + tan ²\(\ \theta\) = sec ²\(\ \theta\); 1 + cot ²\(\ \theta\) = cosec²\(\ \theta\).

They help students quickly recall stan dard values essential for solving exam problems.

Usin g calculators, ignoring diagrams, and misidentifying opposite/adjacent sides.

Visualizing the triangle reduces mistakes and clarifies angle–side relationships.

Understan d the geometric meaning rather than memorizing formulas blindly.

Identify the angle first; the side directly across it is opposite, and the side touching it (except hypotenuse) is adjacent.

Yes, because all angles considered (0\(^\circ\)–90\(^\circ\)) lie in the first quadrant.

It leads to understan ding functions, calculus, vectors, coordinate geometry, and physics waveforms.

sin 30\(^\circ\) = 1/2, derived from the geometry of a 30\(^\circ\)-60\(^\circ\) right triangle.

Yes, cartography uses trigonometric principles to estimate distan ces and directions.

To introduce students to trigonometric ratios, identities, and basic applications in a right triangle.

Stan dard values, complementary angles, identities, and basic height-distan ce problems.

An angle greater than 0\(^\circ\) and less than 90\(^\circ\).

Finding heights usin g the angle of elevation and a known distan ce.

It teaches ratio relationships, spatial interpretation, and analytical problem-solving.

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