CBSE Board Half Yearly Exam Question Paper Class-X Trigonometry 2022
ABC School
CLASS: X
MATHEMATICS
CBSE Board Half Yearly Exam Class-X Trigonometry
TOPIC – CHAPTER -8 – (INTRODUCTION TO TRIGONOMETRY)
SUBTOPICS:-
- Trigonometric ratiosof an acuteangleofarighttriangle.
- Relation between trigonometric ratios.
- Valuesofthe trigonometric ratiosofspecific angles
- Trigonometric identities.
INSTRUCTIONAL AIDS:-
Presentation by screen sharing, offline whiteboard, online whiteboard, You tube links, E lesson, geometrical instruments and graph paper to verify the relation between different ratios.
LEARNING OUTCOMES:-
Each student will be able to
- definetrigonometricratiosof anacute angleof arighttriangle.
- identify and apply relation between various trigonometric ratios.
- applytrigonometricratiosofspecific angleslike0 o30o ,45o .60o ,90o
- prove and apply trigonometric identitieslike
sin2θ+cos2θ=1
1+cot2θ=coesc2θ
1+tan2θ=sec2θ
Block 1
Introduction activity –
1) Recapitulation of perpendicular ,base and hypotenuse in a right angled triangle and naming ofthesidesasshowninthefollowingimage.
Trigonometry isthestudyofrelationshipbetweenthesidesandtheanglesofthetriangle.In class X, we willstudy about the relation between sides and angles ofright angled triangle only.
Refer to the following link to know What is Trigonometry?
2) Students will observe that the ratios of sides of two different right triangles do not vary, if the angles remain the same through the following activity.
Studentswilldrawa triangleABConagraphpaperwhere coordinates ofA,BandC are (2.2),(5,2), (5,5).Similarly, another triangle AEF can be drawn with vertices A(2,2) ,E(7,2)and Fisthe intersection ofACextended andEF whereEF isparallel to BC.
Students can summarise their observation in the following table.
| In ABC | In AEF | |
| P/H | ||
| B/H | ||
| P/B | ||
| H/P | ||
| H/B | ||
| B/P |
Observations- Students can observe that in both the triangles, angles are respectivelysame and ratio of sides like P/H,B/H,P/B etc remains constant.
Lesson development:-As these ratios are constant therefore these 6 ratios are called
trigonometric ratios and have been given 6 different names as explained below.
Refer to the following link to define six trigonometric ratios of an acute angle ofa right triangle
Six trigonometric ratio of an acute angles of a right triangle – Consider triangle ABC, right- angled at B. These ratios are always defined with respect to acute angle ‘A’ or angle ‘C.
Let us look at both cases:
In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six trigonometric ratios with respect to the sides.
| case I | case II |
(i) sine A =  | (i) sine C =  |
(ii) cosine A =  | (ii) cosine C =  |
(iii) tangent A =  | (iii) tangent C =  |
(iv) cosecant A =  | (iv) cosecant C =  |
(v) secant A =  | (v) secant C =  |
(vi) cotangent A =  | (vi) cotangent C =  |
EXERCISE 8.1
Q1.In Δ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :
(i) sin A, cos A (ii) sin C, cos C
2. In Fig find tan P – cot R
4. Given 15 cot A = 8, find sin A and sec A
6. If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that ∠ A
= ∠ B
8. If 3 cot A = 4, check whether
10. In Δ PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
11. State whether the following are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A = 12 /5 for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin θ = 4/ 3 for some angle θ.
HOME WORK ASSIGNMENT EX.
8.1 Q 3,,5,7, 9
Refer to the following link to do more similar questions.
BLOCK 2-
LESSON DEVELOPMENT
TRIGONOMETRIC RATIOS OF SPECIFIC ANGLES-Trigonometric
ratios of specific angles like 0o,30o,45o,60o,90o .can be found geometrically.
Refer to the following links to obtain the T- ratios of specificangles.
The value of sin θ and cos θ can never exceed 1 as hypotenuse is the longest side.therefore,P/H and B/H is always less than 1.
Let us do a question based on T-ratios of specific angles.
Question
Solution-
EXERCISE 8.2
1. Evaluate the following :
HOME WORK ASSIGNMENT EX.
8.2 Q 1 (1),(ii) ,(iii) Q2
ACTIVITY-
1) Hand trick to learn T-ratios of specific angles.
Following questions to be asked to revise the concepts done in the block.
i) Evaluate sin30o. tan 60o
ii) If sin A =cos B and A=30o,the value of B is .
iii) If tan A=cot B and A and B are acute angles. What all values can A and B take?
iv) What happens to the values of cos α as angle α increases from 0° to 90°?
BLOCK 3.
Kindly note that complimentary angles of trigonometric ratios are deleted, therefore Exercise 8.3 is deleted.
LESSON DEVELOPMENT-
1) Refer to the following link to prove and
apply the identities
(a) sin2θ+cos2θ=1
(b) sec2 A – tan2 A = 1 for 0° ≤ A < 90°,
(c) cosec2 A = 1 + cot 2A for 0° < A ≤ 90º.
Let us practice some questions on trigonometric identities.
EXAMPLE 1.
Example
Example 2
SOLUTION :
EXERCISE 8.4
1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
2. Write all the other trigonometric ratios of ∠ A in terms of sec A.
5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined
Refer to the following links for more practice questions:
PRACTICE ASSIGNMENT (EXAMPLAR QUESTIONS)
3.
4
What is the value of k?
Option 1: 30°
Option 2: 60°
Option 3: 45°
Option 4: 90°
Integration With Other subjects
Biology Marine biologists often use trigonometry to establish measurements. For example, to find out how light levels at different depths affect the ability of algae to photosynthesize. Trigonometry is used in finding the distance between celestial bodies.
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