Below are 20 Important Questions and Answers from CBSE Class 10 Mathematics – Chapter 8: Introduction to Trigonometry, prepared according to the CBSE 2026–27 syllabus. The chapter focuses on trigonometric ratios, complementary angles, and basic identities.
1. What is Trigonometry? Why is it important?
Answer:
Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a right-angled triangle. The word “Trigonometry” comes from the Greek words meaning “triangle” and “measure.” It helps us determine unknown sides and angles using specific ratios. Trigonometry is widely used in engineering, astronomy, navigation, architecture, and surveying. In Class 10, students learn six trigonometric ratios: sin, cos, tan, cosec, sec, and cot. These ratios are defined using the sides of a right triangle. Understanding trigonometry develops logical thinking and provides a foundation for higher mathematics and practical applications in real life.
2. Define the six trigonometric ratios.
Answer:
The six trigonometric ratios are defined with respect to an acute angle in a right-angled triangle. Let θ be an acute angle. Then:
- sin θ = Opposite Side / Hypotenuse
- cos θ = Adjacent Side / Hypotenuse
- tan θ = Opposite Side / Adjacent Side
- cosec θ = Hypotenuse / Opposite Side
- sec θ = Hypotenuse / Adjacent Side
- cot θ = Adjacent Side / Opposite Side
These ratios help us find unknown sides and angles in a triangle. The first three ratios are primary ratios, while the other three are their reciprocals. They form the basis of all trigonometric calculations and identities.
3. What are the sides of a right-angled triangle with respect to an angle?
Answer:
In a right-angled triangle, the names of the sides depend on the chosen acute angle. The longest side opposite the right angle is called the hypotenuse. The side directly opposite the selected angle is called the perpendicular (opposite side). The remaining side, which touches the angle but is not the hypotenuse, is called the base (adjacent side). These terms are important because trigonometric ratios are defined using them. Correct identification of these sides is necessary for solving trigonometric problems accurately. Changing the reference angle changes the opposite and adjacent sides but not the hypotenuse.
4. Why do trigonometric ratios remain constant for a given angle?
Answer:
Trigonometric ratios remain constant for a given angle because all right triangles having the same acute angle are similar. Similar triangles have corresponding sides in the same proportion. Therefore, the ratio of opposite side to hypotenuse, adjacent side to hypotenuse, and opposite side to adjacent side remains unchanged. For example, if two right triangles have an angle of 30°, their sin 30°, cos 30°, and tan 30° values will always be the same regardless of their sizes. This property makes trigonometry a powerful mathematical tool for solving practical problems involving distances, heights, and angles.
5. What is the relationship between tan θ and cot θ?
Answer:
Tan θ and cot θ are reciprocal trigonometric ratios. The tangent of an angle is defined as:
tan θ = Opposite Side / Adjacent Side
The cotangent of the same angle is:
cot θ = Adjacent Side / Opposite Side
Therefore,
tan θ × cot θ = 1
This means that if one ratio is known, the other can easily be found by taking its reciprocal. For example, if tan θ = 5/3, then cot θ = 3/5. This relationship is frequently used in solving trigonometric problems and proving identities. Understanding reciprocal ratios helps students simplify complex expressions quickly and accurately.
6. Explain the relationship between sin θ and cosec θ.
Answer:
Sine and cosecant are reciprocal trigonometric ratios. The sine of an angle is defined as:
sin θ = Opposite Side / Hypotenuse
The cosecant of the same angle is:
cosec θ = Hypotenuse / Opposite Side
Hence,
sin θ × cosec θ = 1
This reciprocal relationship allows us to find one ratio when the other is known. For instance, if sin θ = 4/5, then cosec θ = 5/4. Such relationships are useful in simplifying trigonometric expressions and solving numerical problems. Reciprocal ratios form an important part of trigonometry and are often used in proving trigonometric identities.
7. What is the relationship between cos θ and sec θ?
Answer:
Cosine and secant are reciprocal trigonometric ratios. Cosine is defined as:
cos θ = Adjacent Side / Hypotenuse
Secant is defined as:
sec θ = Hypotenuse / Adjacent Side
Therefore,
cos θ × sec θ = 1
This relationship means that secant can be obtained by taking the reciprocal of cosine and vice versa. For example, if cos θ = 3/5, then sec θ = 5/3. These reciprocal relationships help students solve questions involving unknown ratios and simplify trigonometric expressions. They also play an important role in proving identities and understanding the connections among different trigonometric functions.
8. What are complementary angles?
Answer:
Two angles whose sum is 90° are called complementary angles. For example, 30° and 60° are complementary because their sum is 90°. In trigonometry, complementary angles have special relationships among their ratios. These relationships help simplify calculations and solve problems quickly. The concept is based on the properties of right-angled triangles. Understanding complementary angles is important because many trigonometric identities are derived from them. Questions related to complementary angles are frequently asked in examinations and often require students to apply standard trigonometric relationships accurately.
9. State the trigonometric ratios of complementary angles.
Answer:
The trigonometric ratios of complementary angles are:
- sin (90° − θ) = cos θ
- cos (90° − θ) = sin θ
- tan (90° − θ) = cot θ
- cot (90° − θ) = tan θ
- sec (90° − θ) = cosec θ
- cosec (90° − θ) = sec θ
These identities are based on the properties of a right-angled triangle. They help convert one trigonometric ratio into another and simplify calculations. Such relations are very important in solving numerical problems and proving identities. Students should memorize these formulas carefully because they are frequently used in board examinations.
10. What is a trigonometric identity?
Answer:
A trigonometric identity is an equation involving trigonometric ratios that remains true for all permissible values of an angle. Unlike ordinary equations, identities are always valid. For example:
sin²θ + cos²θ = 1
is a trigonometric identity because it holds true for every angle where the ratios are defined. Identities are used to simplify expressions and verify mathematical statements. They form an important part of trigonometry and provide the basis for solving advanced problems. Students should understand the meaning of identities and learn the fundamental identities thoroughly for board examinations and higher mathematical studies.
11. State the fundamental trigonometric identity.
Answer:
The most important trigonometric identity is:
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This identity is derived from the Pythagoras theorem in a right-angled triangle. It is valid for all angles where the trigonometric ratios are defined. The identity helps in simplifying expressions and proving other trigonometric relationships. Many board examination questions are based on this formula. Students should memorize it and practice applying it in different situations. It is considered the foundation of trigonometric identities and is frequently used in higher mathematics.
12. How can other trigonometric ratios be found if one ratio is known?
Answer:
If one trigonometric ratio of an acute angle is known, the other ratios can be determined using a right-angled triangle and the Pythagoras theorem. For example, if sin θ = 3/5, then opposite side = 3 and hypotenuse = 5. Using Pythagoras theorem, the adjacent side becomes 4. Therefore:
cos θ = 4/5
tan θ = 3/4
cosec θ = 5/3
sec θ = 5/4
cot θ = 4/3
This method is widely used in trigonometric calculations. It helps students find all six ratios accurately and strengthens their understanding of the relationships among different trigonometric functions.
13. Why is the hypotenuse always the longest side?
Answer:
The hypotenuse is the side opposite the right angle in a right-angled triangle. According to the Pythagoras theorem:
Hypotenuse² = Base² + Perpendicular²
Since the square of the hypotenuse equals the sum of the squares of the other two sides, its length must be greater than either side individually. Therefore, the hypotenuse is always the longest side of a right triangle. It plays a crucial role in defining trigonometric ratios such as sine, cosine, cosecant, and secant. Correct identification of the hypotenuse is essential for solving trigonometric problems accurately and avoiding mistakes in calculations.
14. What is the significance of trigonometric identities?
Answer:
Trigonometric identities help simplify complicated expressions and verify mathematical statements. They establish relationships among different trigonometric ratios and reduce lengthy calculations. For example, identities such as sin²θ + cos²θ = 1 allow students to replace one ratio with another and simplify equations. These identities are useful in geometry, physics, engineering, and higher mathematics. In Class 10, understanding identities is important because many board questions involve proving or applying them. Regular practice of identity-based questions improves problem-solving skills and helps students develop confidence in handling trigonometric expressions.
15. Explain the term “acute angle” in trigonometry.
Answer:
An acute angle is an angle whose measure is greater than 0° and less than 90°. In Class 10 trigonometry, all trigonometric ratios are defined for acute angles of a right-angled triangle. The ratios sin, cos, tan, cosec, sec, and cot are calculated using the sides of the triangle corresponding to the chosen acute angle. As the angle changes, the values of these ratios also change. Understanding acute angles is important because the introductory concepts of trigonometry are based entirely on them. Mastery of acute-angle trigonometry prepares students for advanced topics studied in higher classes.
16. Differentiate between sin θ and cos θ.
Answer:
Sin θ and cos θ are two fundamental trigonometric ratios. Their definitions are:
sin θ = Opposite Side / Hypotenuse
cos θ = Adjacent Side / Hypotenuse
The numerator changes while the denominator remains the hypotenuse in both cases. Sin θ measures the relationship between the opposite side and hypotenuse, whereas cos θ measures the relationship between the adjacent side and hypotenuse. Both ratios are essential for solving right-triangle problems. They are also related through the identity sin²θ + cos²θ = 1. Understanding the difference between them is important for solving numerical questions correctly and avoiding confusion during examinations.
17. Why are reciprocal ratios important in trigonometry?
Answer:
Reciprocal ratios provide alternative ways of expressing trigonometric relationships. They are:
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
These ratios simplify calculations and help solve problems where the primary ratios are difficult to use directly. Reciprocal ratios are often used in proving identities and transforming expressions into simpler forms. They also help establish connections among different trigonometric functions. A strong understanding of reciprocal relationships improves computational speed and accuracy. Questions involving reciprocal ratios frequently appear in board examinations and competitive tests.
18. How is trigonometry useful in daily life?
Answer:
Trigonometry has many practical applications in everyday life. It is used to calculate heights of buildings, towers, trees, and mountains without directly measuring them. Engineers use trigonometry while designing bridges, roads, and structures. Surveyors use it for land measurement and mapping. Pilots and navigators use trigonometric principles to determine positions and directions. Astronomers apply trigonometry to study distances between celestial bodies. Even modern technologies such as GPS systems rely on trigonometric calculations. Thus, trigonometry is not only an academic topic but also an essential tool in science, engineering, and daily problem-solving.
19. What precautions should be taken while solving trigonometric problems?
Answer:
While solving trigonometric problems, students should first identify the right angle and correctly determine the hypotenuse, opposite side, and adjacent side. They must choose the appropriate trigonometric ratio based on the information given. Care should be taken while applying identities and reciprocal relationships. Students should avoid confusing complementary-angle formulas and ensure that calculations are performed accurately. Drawing a neat diagram whenever possible helps in understanding the problem better. Regular practice and careful substitution of values reduce errors. Following these precautions improves accuracy and helps students score better marks in examinations.
20. Why is Introduction to Trigonometry considered a foundation chapter?
Answer:
Introduction to Trigonometry is considered a foundation chapter because it introduces the basic concepts, ratios, identities, and relationships used in advanced mathematics. Topics such as heights and distances, coordinate geometry, calculus, physics, and engineering depend heavily on trigonometric principles. This chapter develops analytical and logical thinking by teaching students how to relate angles and sides mathematically. It also provides practical problem-solving skills applicable in real-life situations. A strong understanding of trigonometric ratios, complementary angles, and identities helps students perform well in higher classes and competitive examinations. Therefore, this chapter forms the basis of future mathematical learning.