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Introduction to Trigonometry
1.0Master Right-Angled Ratios, Identity Transformations, and Complementary Angle Bounds in Minutes
Unlock the mathematical code governing angles and side lengths. Learn how to map functions across right-angled vertices, master the constant relationships of reciprocal parameters, and use foundational Pythagorean identities to ace your Class 10 board exams.
2.0Learning Outcomes
After completing this chapter, you will be able to:
- Understand the concept of trigonometric ratios.
- Define sine, cosine, tangent, cosecant, secant, and cotangent.
- Calculate trigonometric ratios for right-angled triangles.
- Remember the values of trigonometric ratios for standard angles.
- Apply trigonometric identities to simplify expressions.
- Understand complementary angle relationships.
- Solve problems involving trigonometric ratios and identities.
- Solve NCERT, competency-based, and CBSE Board examination questions confidently.
3.0Introduction to Trigonometry
Trigonometry is one of the most important branches of mathematics that studies the relationship between the sides and angles of a right-angled triangle. It forms the foundation for higher mathematics and has numerous real-life applications in engineering, architecture, astronomy, navigation, surveying, and physics.
In this chapter, you'll learn about the six trigonometric ratios, their values for standard angles, important trigonometric identities, complementary angles, and methods to solve problems involving right-angled triangles. Mastering these concepts will prepare you for both board examinations and advanced mathematical studies.
The word trigonometry is derived from the Greek words 'tri' (Meaning three), 'gon' (Meaning sides) and 'metron' (meaning measure). Infact trigonometry is the study of relationships between the sides and angles of a triangle. Let us take some examples from our surroundings where right triangles can be imagined to be formed.
Suppose the students of a school are visiting Statue of Unity. Now if a student is looking at the top of the tower, a right-angle triangle can be imagined to be made, as shown in fig.
Finding ratios of side lengths in similar right triangles. Work with a partner. You will need.
- Graph paper
- A protractor
Exploring the concept
- Draw a segment along a horizontal grid line on your graph paper. Label one end point .
- Use a protractor to draw a angle at A.
- Draw segments along vertical grid lines to form right triangles, such as and shown in figure.
Drawing conclusions
- Why are and similar triangles?
- Cut a strip of graph paper to use as a ruler. Use your ruler to find the value of following ratio for and for :
What do you notice? 3. Make a conjecture about the value of the ratio for any triangle similar to to test your conjecture, find the value of this ratio for [i.e. perpendicular (p), Base(b) and hypotenuse(h)]. In the Exploration you found that corresponding ratios of side lengths in similar right triangles do not depend on the lengths of the sides. These ratios depend only on the shape of the triangles as determined by the measures of the acute angles. These constant ratios are so important that they are given names as sine, cosine, tangent, secant, cosecant, cotangent. Here side opposite to angle A is perpendicular; side adjacent to angle A is base & side opposite to right angle is hypotenuse.
4.0Trigonometric ratios
Sine and Cosine of an angle
In right , the sine of , which is written " ", is given by
Numerical Ability 1 In fig., for , find and .
Building Concepts 1 Between A.D. 1000 and 1300, the Anasazi people lived in cliff dwellings in the southwestern part of the United States. The doors to the cliff dwellings opened onto balconies that were reached by climbing ladders. Suppose the ladder shown rests on the ground and extends above the balcony. How long is the ladder? (use ) Explanation: Use the sine ratio to find the unknown side length.
Building Concepts 2 Indicate the perpendicular, the hypotenuse and the base (in that order) with respect to the angle marked .
5.0Tangent of an angle
In addition to the sine and cosine ratios, you can use the tangent ratio to find the measures of the sides and angles of a right triangle.
In right , the tangent of , which is written "tan ", is given by
Building Concepts 3 Surveying surveyors use trigonometry to calculate distances that would otherwise be difficult to find, such as the distance between two houses located across a lake from each other, as shown in the diagram. What is this distance? (Use ) Explanation Let the distance in meters across the lake. Write an equation involving x and a trigonometric ratio. The distance between the two houses is about 48 m .
Cosecant, secant and cotangent of an Angle
- The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA. Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse Tangent Opposite Adjacent The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Some Officers Have Curly Auburn Hair Till Old Age".
Numerical Ability 2 Using the information given in figure. write the values of all trigonometric ratios of angle .
- Pythagoras theorem : In a right angle triangle, Hypotenuse Perpendicular Base
6.0Reciprocal Relations
Clearly, we have : (i) (ii) (iii)
Thus, we have : (i) (ii) (iii)
7.0Quotient Relations
Consider a right angled triangle in which for an acute angle , we have : Now, (by def.) and (by def.) Thus, and
Numerical Ability 3 In a right , if is acute and , find the remaining trigonometric ratios of . Solution: Consider a in which . For , we have : Base , perpendicular and Hypotenuse .
8.0Power of T-ratios
We denote: (i) by (ii) by (iii) by (iv) by and so on.
- The symbol is used as an abbreviation for 'the sine of the angle '. is not the product of 'sin' and A. 'sin' separated from A has no meaning. Similarly, A is not the product of 'cos' and A. Similar interpretations follow for other trigonometric ratios also.
- We may write , etc., in place of , , etc., respectively. But (it is called sine inverse ). has a different meaning, which will be discussed in higher classes. Similar conventions hold for the other trigonometric ratios as well.
- Since the hypotenuse is the longest side in a right triangle, the value of or is always less than 1 (or, in particular, equal to 1 ).
Numerical Ability 4 If , verify that Solution:
9.0Trigonometric ratio of standard angles
T-ratios of 45^{\circ}
Consider a in which and . Then, clearly, .
T-ratios of \mathbf{6 0} and \mathbf{3 0 ^ { \circ }}
Draw an equilateral with each side . Then, . From A, draw . Then, and .
T-ratios of 60^{\circ}
In we have: and . Base , Perpendicular and Hypotenuse .
T-ratios of \mathbf{3 0}{ }^{\circ}
In we have: and . Base , Perpendicular and Hypotenuse . ; Table for T-ratios of standard angles
Quick Tips (i) As increases from to increases from 0 to 1 . (ii) As increases from to decreases from 1 to 0 . (iii) As increases from to increases from 0 to . (iv) The maximum value of is one. (v) As decreases from 1 to increases from to . (vi) and can not be greater than one numerically. (vii) and can not be less than one numerically. (viii) and can have any value.
Numerical Ability 5 In , right angled at , find the length of the sides and AC. Solution: We are given , i.e., and Now, or or or or 10 cm
Numerical Ability 6 In , right angled at , if and . Find and . Solution
Numerical Ability 7 Find the value of in each of the following: (i) (ii) (iii) Solution: (i) we have, (ii) we have, (iii) we have,
10.0T-ratios of complementary angles
Complementary angles
Two angles are said to be complementary, if their sum is . Thus, and are complementary angles.
T-ratios of complementary angles Consider in which and .
Let and . Then, and . When we consider the T-ratios of , then Base , Perpendicular and Hypotenuse .
$\therefore \quad \sin \left(90^{\circ}-\theta\right)=\frac{A B}{A C}=\frac{x}{r}=\cos \theta \ \cos \left(90^{\circ}-\theta\right)=\frac{\mathrm{BC}}{\mathrm{AC}}=\frac{\mathrm{y}}{\mathrm{r}}=\sin \theta . \ \tan \left(90^{\circ}-\theta\right)=\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{x}{y}=\cot \theta \
\therefore \quad \operatorname{cosec}\left(90^{\circ}-\theta\right)=\frac{1}{\sin \left(90^{\circ}-\theta\right)}=\frac{1}{\cos \theta}=\sec \theta \ \sec \left(90^{\circ}-\theta\right)=\frac{1}{\cos \left(90^{\circ}-\theta\right)}=\frac{1}{\sin \theta}=\operatorname{cosec} \theta \ \cot \left(90^{\circ}-\theta\right)=\frac{1}{\tan \left(90^{\circ}-\theta\right)}=\frac{1}{\cot \theta}=\tan \theta$
(i) (ii) (iii) (iv) (v) (vi)
In other words: angle complement (angle) complement (angle) (complement); (angle) (complement) angle complement (angle) (complement) where complement angle
Numerical Ability 8 Without using tables, evaluate: (i) (ii) (iii) Solution (i) (ii) (iii)
- The above example suggests that out of the two t-ratios, we convert one in term of the t-ratio of the complement.
- For uniformity, we usually convert the angle greater than in terms of its complement.
Numerical Ability 9 Without using tables, show that . Solution:
Numerical Ability 10 Express in terms of trigonometric ratios of angles between and . Solution: .
Numerical Ability 11 Evaluate : Solution: . .
11.0Trigonometric identities
We know that an equation is called an identity when it is true for all values of the variables involved. Similarly, "an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved."
The three Fundamental Trigonometric Identities are - (i) (ii) (iii)
Geometrical proof
Consider a , right angled at . Then we have : By Pythagoras theorem (i) Dividing each term of (i) by , we get i.e., i.e., This is true for all A such that So, this is a trigonometric identity. (ii) Let us now divide (i) by . We get i.e., This equation is true for . Since and sec are not defined for , so (iii) is true for all A such that (iii) Again, let us divide (i) by , we get Since and are not defined for , therefore (iv) is true for all A such that
Using the above trigonometric identities, we can express each trigonometric ratio in terms of the other trigonometric ratios, i.e., if any one of the ratios is known, we can also determine the values of other trigonometric ratios.
Fundamental identities (results)
To prove trigonometrical identities
The following methods are to be followed: Method-I: Take the more complicated side of the identity (L.H.S. or R.H.S. as the case may be) and by using suitable trigonometric and algebraic formulae prove it equal to the other side. Method-II: When neither side of the identity is in a simple form, simplify the L.H.S. and R.H.S. separately by using suitable formulae (by expressing all the T-ratios occurring in the identity in terms of the sine and cosine and show that the results are equal).
Numerical Ability 12 Prove Solution:
Numerical Ability 13 Prove that : Solution:
Numerical Ability 14 Prove that : Solution: Hence Proved.
Numerical Ability 15 Prove the following by using suitable identities: Solution:
Hence Proved.
- In proving question, generally we convert all the trigonometric ratios in terms of and to make it easy and convenient.
Numerical Ability 16 Prove that Solution: Consider the numerator of the LHS of the given expression. i.e. [Use ] [Use ] Take as common] [Note this step very carefully] ... (1) [Rearranging] LHS
12.0Important topics in Class 10 Maths: Trigonometry
13.0EUREKA by ALLEN – Your Smart Companion for Class 10 Success
Designed for all Class 10 students preparing for their CBSE & State Board Exam, ALLEN EUREKA is a unique digital program utilizing very specific learning techniques to enable students to master their content. It brings together real experience with professional content to ensure that students can understand and recall difficult subject areas. In addition, EUREKA provides opportunities for students to develop strong fundamental skills and boost their confidence when doing exam-focused work through the use of interactive lessons, AI doubt-clearing tools, and elaborate practice opportunities.
14.0Supporting Study Materials
This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 9 of Class 10 Maths, is aligned with the latest NCERT guidelines. Complete with side-by-side ratio memory tables, standard value lookups, and clear identity transformation tracks, this guide ensures complete confidence for your board examinations.
30-Second Quick Review: Trigonometry
- There are six trigonometric ratios: sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ
✔ Reciprocal Ratios:
- cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
✔ Fundamental Identity: sin²θ + cos²θ = 1
✔ Complementary Angles:
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
✔ Standard Values:
- sin30° = 1/2
- cos60° = 1/2
- tan45° = 1
✔ Trigonometric ratios are defined only for right-angled triangles in this chapter.
15.0Previous Year Questions (PYQs) on Trigonometry
A surveyor observes the top of a tree using a clinometer. The angle formed with the horizontal is 60°, and the distance from the observation point to the tree is 20 m.
(i) Which trigonometric ratio is most suitable for finding the height?
Answer: Tangent Ratio
(ii) What is the height of the tree?
Height = 20 × tan60°
= 20√3 m
(iii) Which side represents the adjacent side?
Answer: Horizontal distance from the observer to the tree.
(iv) Which branch of mathematics is applied in this situation?
Answer: Trigonometry
16.0Recommended Next Topics
- Circles
- Areas Related to Circles
- Surface Areas and Volumes
- Statistics
- Probability