Test 08 || Class 10th Maths (1) || Ch. 08 Introduction to Trigonometry

Test 08 || Class 10th Maths (1) || Ch. 08 Introduction to Trigonometry

Trigonometric Ratios

Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

Let ∆ABC be a triangle right angled at B. Then the trigonometric ratios of the angle A in right ∆ABC are defined as follows:

Exercise 8.1

Trigonometry

Trigonometry is the study of relationships between the sides and angles of a right-angled triangle.

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
Solution:

by Pythagorean Theorem

H² = P² + B² 

      = (24)² + (7)²

      = 576 +49

H²  = 25²

H= 25

Question 2.
In given figure, find tan P – cot R.
Solution:

by Pythagorean Theorem

Question 3.

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

Question 4.
Given 15 cot A = 8, find sin A and sec A.
Solution:

Question 5.
Given sec θ = 13/12 , calculate all other trigonometric ratios.
Solution:

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

Question 6.
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
Solution:

Angle opposite to equal sides are equal.

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

(ii) cot²θ
Solution:

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

Question 8.

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

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
Solution:

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

Question 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.
Solution:

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

Question 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.
Solution:

(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 

Note:– The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains same.

Question 1.
Evaluate the following:

(i) sin 60° cos 30° + sin 30° cos 60°

(ii) 2tan²45°+ cos²30°– sin²30° 

(iii) 

           cos 45°              

   sec 30° + cosec 30°

(iv) 

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

(iv) 

   5cos²30° + 4sec30°– tan45°                 sin²30° + cos²60°

Solution:

Question 2.
Choose the correct option and justify your choice:

2 tan30°/(1+tan² 30°)

Solution:

Ex 8.2 Class 10 Maths 

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

Question 4.
State whether the following statements are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B.
(ii) The value of sin θ increases as θ increases.
(iii) The value of cos θ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.
Solution:

 Ex 8.3 

Trigonometric Ratios for Complementary Angles

sin (90° – A) = cos A
cos (90° – A) = sin A
tan (90° – A) = cot A
cot (90° – A) = tan A
sec (90° – A) = cosec A
cosec (90° – A) = sec A
Note:
Here (90° – A) is the complementary angle of A.

Question 1.
Evaluate:

Solution:

Question 2.
Show that:
(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0
Solution:

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

Question 4.
If tan A = cot B, prove that A + B = 90°.
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+C2) = cos A2
Solution:

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

Ex 8.4 

Trigonometric Identities

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.[for 0° ≤ θ ≤ 90°]

(i) sin²θ + cos²θ = 1 

(ii) sec²θ – tan²θ = 1 

(iii) cosec²θ – cot²θ = 1 

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+tan²A)/(1+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: