Given that, cos 195 ° Now, cos 195 ° = cos ( 180 ° + 15 ° ) We know that cos 180 ° + θ = - cos θ So, cos 195 ° = - cos 15 ° = - cos ( 45 ° - 30 ° ) = - cos 45 ° cos 30 ° + sin 45 ° sin 30 ° ∵ cos ( A - B ) = cos A cos B + sin A sin B = - 1 2 × 3 2 - 1 2 × 1 2 ∴ cos 195 ° = - 3 + 1 2 2

Evaluate sin 11 π 12 .

Given, sin 11 π 12 = sin π - π 12 = sin π 12 ∵ sin ( π - θ ) = sinθ = sin π 3 - π 4 = sin π 3 cos π 4 - sin π 4 cos π 3 ∵ sin ( A - B ) = sin A cos B - sin B cos A = 3 2 · 1 2 - 1 2 · 1 2 = 3 - 1 2 2 ∴ sin 11 π 12 = 3 - 1 2 2 .

If tan A = m tan B , prove that sin ( A - B ) sin ( A + B ) = m - 1 m + 1 .

Given that, tan A = m tan B LHS = sin ( A - B ) sin ( A + B ) = sin A cos B - sin B cos A sin A cos B + sin B cos A = sin A cos B - sin B cos A cos A cos B sin A cos B + sin B cos A cos A cos B = tan A - tan B tan A + tan B = m tan B - tan B m tan B + tan B = m - 1 m + 1 · tan B tan B = m - 1 m + 1 = RHS Hence, proved that sin ( A - B ) sin ( A + B ) = m - 1 m + 1 .

If α + β = γ and tan α tan β = x y , then prove that sin ( α - β ) = x - y x + y sin γ .

Given that, tan α tan β = x y Apply componendo and dividendo i.e., a b = c d implies a ± b a ∓ b = c ± d c ∓ d . = tan α + tan β tan α - tan β = x + y x - y ⇒ sin α cos α + sin β cos β sin α cos α - sin β cos β = x + y x - y ⇒ sin α cos β + sin β cos α sin α cos β - sin β cos α = x + y x - y ⇒ sin ( α + β ) sin ( α - β ) = x + y x - y ⇒ sin ( α - β ) = x - y x + y sin α + β ⇒ sin ( α - β ) = x - y x + y sin γ Hence, proved that sin ( α - β ) = x - y x + y sin γ .

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Understanding ISC Mathematics Class 11 Volume 1>Chapter 3 - Trigonometry>EXERCISE 3.5

M L Aggarwal Solutions for Chapter: Trigonometry, Exercise 5: EXERCISE 3.5

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Trigonometry, Exercise 5: EXERCISE 3.5

Attempt the practice questions on Chapter 3: Trigonometry, Exercise 5: EXERCISE 3.5 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.