Class 10 Maths

# Trigonometry

## NCERT Exercise 8.3

Question – 1 - Evaluate the following:

(i) (text(sin)18°)/(text(cos)72°)

(text(sin)18°)/(text(cos)72°)

=(text(sin)(90°-72°))/(text(cos)72°)

=(text(cos)72°)/(text(cos)72°)=1

(ii) (text(tan)26°)/(text(cot)64°)

(text(tan)26°)/(text(cot)64°)

=(text(tan)(90°-64°))/(text(cot)64°)

=(text(cot)64°)/(text(cot)64°)=1

(iii) cos 48o - sin 42o

Answer: cos 48o - sin 42o

= cos (90o - 42o) – sin 42o

= sin 42o - sin 42o = 0

(iv) cosec 31o - sec 59o

Answer: cosec 31o - sec 59o

= cosec(90o - 59o) – sec 59o

= sec 59o - sec 59o = 0

Question – 2 - Show that:

(i) tan 48o tan 23o tan 42o tan 67o = 1

Answer: LHS: tan 48o tan 23o tan 42o tan 67o

= tan (90o - 42o) x tan 23o x tan 42o x tan (90o - 23o)

= cot 42o x tan 23o x tan 42o x cot 23o

= cot 42o x tan 42o x tan 23o x cot 23o

= 1 x 1 = 1 Proved

(ii) cos 38o cos 52o - sin 38o sin 52o = 0

Answer: cos 38o cos 52o - sin 38o sin 52o = 0

= cos (90o - 52o) cos (90o - 38o) - sin 38o sin 52o

= sin 38o sin 52o - sin 38o sin 52o = 0 proved

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

Answer: tan 2A = cot (90° – 2A)

This means;

90° – 2A = A – 18°

Or, 108° – 2A = A

Or, 3A = 108°

Or, A = 108/3 = 36°

Question – 4 - If tan A = cot B, prove that A + B = 90°

Answer: tan A = cot (90°- A) = cot B

This means, 90° - A = B

Or, A + B = 90° proved.

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

Answer: sec 4A = cosec (90° - 4A) = cosec (A - 20°)

This means; 90° – 4A = A - 20°

Or, 110°– 4A = A

Or, 5A = 110°

Or, A = 22°

Question – 6 - If A, B and C are interior angles of a triangle ABC, then show that

text(sin)((B+C)/(2))=text(cos)A/2

Answer: Since, A + B + C = 90°

So, B + C = 90°- A

Hence,

text(sin)((B+C)/(2))=text(sin)((90°-A)/(2))

=text(cos)A/2 proved

Question – 7 - Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

Answer: sin 67° + cos 75° can be written as follows:

Sin (90° – 23°) + cos (90° – 15°)

= cos 23° + sin 15°