Class 10 || Maths || Exe || Ch. 08.01 Introduction to Trigonometry

Class 10 || Maths || Exe || Ch. 08.01 Introduction to Trigonometry  (त्रिकोणमिति)

Q. 1.
In ∆ABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C
Sol.

by Pythagorean Theorem

H² = P² + B² 

      = (24)² + (7)²

      = 576 +49

H²  = 25²

H= 25

Q. 2. In given figure, find tan P – cot R.
Sol.

by Pythagorean Theorem

Q. 3. If sin A = 3/4 , calculate cos A and tan A.

Sol.

Q. 4. Given 15 cot A = 8, find sin A and sec A.
Sol.

Q. 5. Given sec θ = 13/12 , calculate all other trigonometric ratios.
Sol.

We know that 

sec θ = 13/12 

AB/AC = 13/12

Let AC = 13k and AB = 12k

by the Pythagorean Theorem

AC² = AB² + BC²  

BC² = AC² – AB²  = (13k)² –(12k)²

= 169k² – 144k² = 25k²

BC² = 25k²

BC = 5k

Q. 6. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
Sol.

Angle opposite to equal sides are equal.

Q. 7. If cot θ = 7/8, evaluate:
(i) [(1+sinθ)(1−sinθ)][ / [(1+cosθ)(1−cosθ)]

(ii) cot²θ
Sol.

We know that 

cot θ = 7/8 

AB/BC = 7/8

let AB = 7k and BC =8k 

then in ∆ABC

AC² = AB² + BC²  

= (7k)² + (8k)²

= 49k² + 64k² = 113k²

AC² = 113k²

AC = k√113

Q. 8. If 3 cot A = 4, check whether (1−tan²A)/(1+tan²A) = cos² A – sin² A or not.

Sol.

do yourself

 Q. 9. I Intriangle 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
Sol.

We know that 

tan θ = 1/√3 

BC/AB = 1/√3 

let BC = 1k and AB =√3k 

then in ∆ABC

AC² = AB² + BC²  

= (√3k)² + (1k)²

= 3k² + 1k² = 4k²

AC² = 4k²

AC = 2k

(i) sin A cos C + cos A sin C

(ii) cos A cos C – sin A sin C

Q. 10. In ΔPQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
Sol.

We know that 

PR + QR = 25 cm

PQ = 5 cm.

Let QR = x 

then PR = 25 – x

by the Pythagorean Theorem

PR² = PQ² + QR²  

PQ² = PR² – QR² 

(5)² = (25–x )² – (x)² 

25 = (25)² + (x)² – 2×25×x –(x)²

25 = 625 + x² – 50x –x²

25 = 625  – 50x

50x = 625–25

50x = 600

x= 600/50=12

Now PR = 25 – 12

PR = 13

Q. 11. State whether the following statements 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.
Sol.

(i) False

the value depends on the sides of the triangle so then we may have any value.

(ii) True

Because sec A is always greater than 1

(iii) False

cos A is the abbreviation used for the cosine of angle A.

(iv) False

cot A is not the product of cot and A it is single cot A.

(v) False

The value of sin theta cannot be less than 1

Exercise 8.2 

Question 1.
Evaluate the following:

   sin 30° + tan 45°–cosec60°       sec 30° + cos 60° + cot 45°

Solution:

Question 3.
If tan (A + B) = √3 and tan (A – B) = 1√3; 0° < A + B ≤ 90°; A > B, find A and B.
Solution:

 Ex 8.3 

Question 1.
Evaluate:

Solution:

Question 3.
If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.
Solution:

Question 5.
If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.
Solution:

Question 6.
If A, B and C are interior angles of a triangle ABC, then show that: sin (B+C)/2 = cos A/2
Solution:

Question 7.
Express sin 61° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.
Solution:

Ex 8.4 

Question 1.

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.Solution:

Question 2.
Write all the other trigonometric ratios of ∠A in terms of sec A.
Solution:
 

Question 3.

Evaluate:

Solution:

Question 4.
Choose the correct option. Justify your choice.
(i) 9 sec² A – 9 tan² A = ……
(A) 1                 (B) 9
(C) 8                 (D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) = ………..
(A) 0                (B) 1
(C) 2                (D) -1

(iii) (sec A + tan A) (1 – sin A) = ………….
(A) sec A            (B) sin A
(C) cosec A        (D) cos A

(iv) 1+tan2A1+cot2A = ………..
(A) sec² A.          (B) -1
(C) cot² A.           (D) tan² A
Solution:

Question 5.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Solution:

A