Exercise 8.1 (NCERT) - Trigonometry, Class 10
| Chapter 8 | Exercise 8.1 | Class 10 |
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Below are the Quick links for all questions of Exercise 8.1, Trigonometry, Class 10.
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Below are the solutions of every question of exercise 8.1 of chapter 8, Introduction to Trigonometry, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.
Question 1
In △ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine : (i) sin A, cos A (ii) sin C, cos C
∵ △ ABC is a right angled triangle, let us apply pythagoras theorem
(i) Let us now find the trigonometric ratios sin A and cos A
When we take A as the angle, the perpendicular will be side opposite to it. So
BC – Perpendicular, AB – Base, AC – Hypotenuse
(ii) Now we have to find the trigonometric ratios sin C and cos C
When we take C as the angle, the perpendicular will be side opposite to it. So
AB – Perpendicular, BC – Base, AC – Hypotenuse
Question 2
In Fig. 8.13, find tan P – cot R.
∵ △ PQR is a right angled triangle, let us apply pythagoras theorem and find QR.
Let us now find the trigonometric ratios tan P and cot R
When we take P as the angle, the perpendicular will be side opposite to it. So
QR – Perpendicular, PQ – Base, PR – Hypotenuse
When we take R as the angle, the perpendicular will be side opposite to it. So
PQ – Perpendicular, QR – Base, PR – Hypotenuse
From equations (1) and (2)
Question 3
If sinA = 3/4 calculate cos A and tan A.
Now we know that
Note: Now this is an important point to understand. Often students take Perpendicular (BC) as 3 and Hypotenuse (AC) as 4. But 3/4 is ratio of Perpendicular and Hypotenuse. Hence using the above ratio, we must solve it as follows.
Now, let us apply Pythagoras theorem to find the third side (AB) of the right angled triangle.
Let us now calculate cos A and tan A
When we take A as the angle, the perpendicular will be the side opposite to it. Hence
BC – Perpendicular, AB – Base, AC – Hypotenuse
Question 4
Given 15 cot A = 8, find sin A and sec A.
This question is similar to the previous question that we just solved.
When we take A as the angle, the perpendicular will be side opposite to it. So
BC – Perpendicular, AB – Base, AC – Hypotenuse
Now we know that
Note: Now this is an important point to understand. Often students take Base (AB) as 8 and Perpendicular (BC) as 15. But 8/15 is ratio of Base and Perpendicular. Hence using the above ratio, we must solve it as follows.
Now, let us apply Pythagoras theorem to find the third side (AC) of the right angled triangle.
Let us now calculate sin A and sec A
Question 5
Given sec θ = 13/12 calculate all other trigonometric ratios.
Super Tip: We have to find all other trigonometry ratios in this question and for that we need to know the dimensions of all 3 sides of the triangle. But before that, we can directly find the value of cos θ. This way, you will quickly score 20% marks 😊
cos θ
Let us consider that angle θ is at vertex A of right angled triangle ABC with right angle at B. This means
BC – Perpendicular, AB – Base, AC – Hypotenuse
Note: Now this is an important point to understand. Often students take Hypotenuse (AC) as 13 and Base (AB) as 12. But 13/12 is ratio of Hypotenuse and Base. Hence using the above ratio, we must solve it as follows.
Now, let us apply Pythagoras theorem to find the third side (BC) of the right angled triangle.
sin θ
cosec θ
Now cosec θ can be calculated in 2 ways. (Write only 1 of these methods in exam)
Method 1
Method 2
tan θ
Now tan θ can be calculated in 2 ways. (Write only 1 of these methods in exam)
Method 1
Method 2
cot θ
Now cot θ can be calculated in 2 ways. (Write only 1 of these methods in exam)
Method 1
Method 2
Question 6
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
It is given that ∠A and ∠B are acute angles
Now for angle A
Perpendicular = BC (side opp. to vertex A)
Base = AC (side adjacent to vertex A)
& for angle B
Perpendicular = AC (side opp. to vertex B)
Base = BC (side opposite to vertex B)
Hypotenuse (AB) will remain same as hypotenuse in a right angled triangle is the longest side.
Hence
But we know that
cosA = cosB
From (1) and (2)
This implies △ABC is an isosceles triangle.
Question 7
If cot θ = 7/8 evaluate :
Time saver tip: Such questions are very easy to solve. No need to find any other trigonometric ratio to solve this question.
Question 8
If 3 cot A = 4, check whether
When we take A as the angle, then
BC – Perpendicular, AB – Base, AC – Hypotenuse
Now we know that
Note: Now this is an important point to understand. Often students take Base (AB) as 4 and Perpendicular (BC) as 3. But 4/3 is ratio of Base and Perpendicular. Hence using the above ratio, we must solve it as follows.
Applying Pythagoras theorem
Let’s calculate the values of sin A, cos A & tan A
Hence proved.
Question 9
In triangle ABC, right-angled at B, if tan A = 1/√3 find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
When we take A as the angle, the perpendicular will be side opposite to it. So
BC – Perpendicular, AB – Base, AC – Hypotenuse
Now we know that
If you have any doubt about the above statement, read the first NOTE written on this page.
Now, using Pythagoras theorem
(i) sin A cos C + cos A sin C
When we take C as the angle, the perpendicular will be side opposite to it. So
AB – Perpendicular, BC – Base, AC – Hypotenuse
(ii) cos A cos C – sin A sin C
Question 10
In △PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
Applying Pythagoras theorem
Let’s calculate the values of sin P, cos P & tan P
Question 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
Solution:
(i) The value of tan A is always less than 1 – False
Reason: We know that tan 60° = √3 and √3 > 1, hence this statement is false.
(i) sec A = 12/5 for some value of angle A. – True
Reason: Trigonometric ratio sec is ratio of Hyp/Base and we know that Hypotenuse is always the longest side of a right angled triangle. Hence taking Hyp = 12 and Base = 5 we can get a right angled triangle.
(iii) cos A is the abbreviation used for the cosecant of angle A – False
Reason: cos A is abbreviation of cosine of angle A.
(iv) cot A is the product of cot and A – False
Reason: cot A is single a term and it is cotangent of A.
(v) sin θ = 4/3 for some angle θ – False
Reason: Value of sin θ can never be greater than 1 because it is the ratio of Perpendicular and Hypotenuse and perpendicular can never be longer than hypotenuse.