Within what respective limits must A 2 lie when 2 sin A 2 = 1 + sin A - 1 - sin A ?

∵ 2 sin A 2 = 1 + sin A - 1 - sin A ⇒ 2 sin A 2 = sin 2 A 2 + cos 2 A 2 + 2 sin A 2 cos A 2 - sin 2 A 2 + cos 2 A 2 - 2 sin A 2 cos A 2 ⇒ 2 sin A 2 = sin A 2 + cos A 2 2 - sin A 2 - cos A 2 2 ⇒ 2 sin A 2 = sin A 2 + cos A 2 - sin A 2 - cos A 2 ∴ sin A 2 + cos A 2 ≥ 0 and sin A 2 - cos A 2 ≤ 0 . Hence, principal solution will be A 2 ∈ - π 4 , π 4 . ⇒ A 2 ∈ 2 n π - π 4 , 2 nπ + π 4 .

Within what respective limits must A 2 lie when 2 cos A 2 = 1 + sin A - 1 - sin A ?

∵ 2 cos A 2 = 1 + sin A - 1 - sin A ⇒ 2 cos A 2 = sin 2 A 2 + cos 2 A 2 + 2 sin A 2 cos A 2 - sin 2 A 2 + cos 2 A 2 - 2 sin A 2 cos A 2 ⇒ 2 cos A 2 = sin A 2 + cos A 2 2 - sin A 2 - cos A 2 2 ⇒ 2 cos A 2 = sin A 2 + cos A 2 - sin A 2 - cos A 2 ∴ sin A 2 + cos A 2 ≥ 0 and sin A 2 - cos A 2 ≥ 0 . Hence, principal solution will be A 2 ∈ π 4 , 3 π 4 . ⇒ A 2 ∈ 2 n π + π 4 , 2 nπ + 3 π 4 , n ∈ I

In the formula 2 cos A 2 = ± 1 + sin A ± 1 - sin A , find within what limits A 2 must lie when the two positive signs are taken.

∵ 2 cos A 2 = 1 + sin A + 1 - sin A ⇒ 2 cos A 2 = sin 2 A 2 + cos 2 A 2 + 2 sin A 2 cos A 2 + sin 2 A 2 + cos 2 A 2 - 2 sin A 2 cos A 2 ⇒ 2 cos A 2 = sin A 2 + cos A 2 2 + sin A 2 - cos A 2 2 ⇒ 2 cos A 2 = sin A 2 + cos A 2 + sin A 2 - cos A 2 ∴ sin A 2 + cos A 2 ≥ 0 and sin A 2 - cos A 2 ≤ 0 . Hence, principal solution will be A 2 ∈ - π 4 , π 4 . ⇒ A 2 ∈ 2 n π - π 4 , 2 nπ + π 4

In the formula 2 cos A 2 = ± 1 + sin A ± 1 - sin A , find within what limits A 2 must lie when the two negative signs are taken?

Given, 2 cos A 2 = - 1 + sin A - 1 - sin A We know, sin 2 θ + cos 2 θ = 1 and sin 2 θ = 2 sinθcosθ ⇒ - 2 cos A 2 = sin 2 A 2 + cos 2 A 2 + 2 sin A 2 cos A 2 + sin 2 A 2 + cos 2 A 2 - 2 sin A 2 cos A 2 ⇒ - 2 cos A 2 = sin A 2 + cos A 2 2 + sin A 2 - cos A 2 2 ⇒ - 2 cos A 2 = sin A 2 + cos A 2 + sin A 2 - cos A 2 The above equation holds true only when sin A 2 + cos A 2 is negative and sin A 2 - cos A 2 is positive. Hence, A 2 must be lie between 135 ° and 225 ° or A 2 must be lie between 2 n π + 3 π 4 and 2 n π + 5 π 4 , where n is an integer.

In the formula 2 cos A 2 = ± 1 + sin A ± 1 - sin A , find within what limits A 2 must lie when the first sign is negative and the second positive?

Given, 2 cos A 2 = - 1 + sin A + 1 - sin A We know, sin 2 θ + cos 2 θ = 1 and sin 2 θ = 2 sinθcosθ ⇒ 2 cos A 2 = - sin 2 A 2 + cos 2 A 2 + 2 sin A 2 cos A 2 + sin 2 A 2 + cos 2 A 2 - 2 sin A 2 cos A 2 ⇒ 2 cos A 2 = - sin A 2 + cos A 2 2 + sin A 2 - cos A 2 2 ⇒ 2 cos A 2 = - sin A 2 + cos A 2 + sin A 2 - cos A 2 The above equation holds true only when both sin A 2 + cos A 2 and sin A 2 - cos A 2 are negative. Hence, A 2 must be lie between 225 ° and 315 ° or A 2 must be lie between 2 n π + 5 π 4 and 2 n π + 7 π 4 , where n is an integer.

If sin A 3 be determined from the equation sin A = 3 sin A 3 - 4 sin 3 A 3 , prove that we should expect to obtain also the value of sin π - A 3 and - sin π + A 3 .

Given that sin A = 3 sin A 3 - 4 sin 3 A 3 … i From equation i , we can find the value of sin A 3 . But sin A = sin π - A and sin π - A = 3 sin π - A 3 - 4 sin 3 π - A 3 … ii It is clear that the equations i & ii are same. So, from equation ii , the value of sin π - A 3 can be found. Again, sin A = sin - π + A and sin - π + A = 3 sin - π + A 3 - 4 sin 3 - π + A 3 … iii It is clear that the equations i & iii are same. So, from equation ( iii ) , the value of - sin π + A 3 can be found. Hence, we can get the value of sin π - A 3 and - sin π + A 3 from the value of sin A .

If cos A 3 be determined from the equation cos A = 4 cos 3 A 3 - 3 cos A 3 , prove that we should expect to obtain also the value of cos 2 π - A 3 and cos 2 π + A 3 .

Given that cos A = 4 cos 3 A 3 - 3 cos A 3 … i From equation i , we can find the value of cos A 3 . But cos A = cos 2 π - A and cos 2 π - A = 4 cos 3 2 π - A 3 - 3 cos 2 π - A 3 … ii It is clear that the equations i & ii are same. So, from equation ii , the value of cos 2 π - A 3 can be found. Again, cos A = cos 2 π + A and cos 2 π + A = 4 cos 3 2 π + A 3 - 3 cos 2 π + A 3 … iii It is clear that the equations i & iii are same. So, from equation iii , the value of cos 2 π + A 3 can be found. Hence, we can get the value of cos 2 π - A 3 and cos 2 π + A 3 from the value of cos A .

E M B I B E

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII

S L Loney Solutions for Chapter: Trigonometrical Ratios of Multiple and Submultiple of Angles, Exercise 2: EXAMPLES XVIII

Author:S L Loney

S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: Trigonometrical Ratios of Multiple and Submultiple of Angles, Exercise 2: EXAMPLES XVIII

Attempt the practice questions on Chapter 8: Trigonometrical Ratios of Multiple and Submultiple of Angles, Exercise 2: EXAMPLES XVIII with hints and solutions to strengthen your understanding. Plane Trigonometry Part 1 solutions are prepared by Experienced Embibe Experts.

Questions from S L Loney Solutions for Chapter: Trigonometrical Ratios of Multiple and Submultiple of Angles, Exercise 2: EXAMPLES XVIII with Hints & Solutions

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 29

Within what respective limits must $\frac{A}{2}$ lie when $2 \sin \frac{A}{2} = \sqrt{1 + \sin A} - \sqrt{1 - \sin A}$ ?

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 29

Within what respective limits must $\frac{A}{2}$ lie when $2 \cos \frac{A}{2} = \sqrt{1 + \sin A} - \sqrt{1 - \sin A}$ ?

EASY

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 30

In the formula $2 \cos \frac{A}{2} = \pm \sqrt{1 + \sin A} \pm \sqrt{1 - \sin A}$ , find within what limits $\frac{A}{2}$ must lie when the two positive signs are taken.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 30

In the formula $2 \cos \frac{A}{2} = \pm \sqrt{1 + \sin A} \pm \sqrt{1 - \sin A}$ , find within what limits $\frac{A}{2}$ must lie when the two negative signs are taken?

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 30

In the formula $2 \cos \frac{A}{2} = \pm \sqrt{1 + \sin A} \pm \sqrt{1 - \sin A}$ , find within what limits $\frac{A}{2}$ must lie when the first sign is negative and the second positive?

EASY

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 31

Prove that the sine is algebraically less than the cosine for any angle between $2 n π - \frac{3 π}{4}$ and $2 n π + \frac{π}{4}$ , where $n$ is any integer.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 32

If $\sin \frac{A}{3}$ be determined from the equation $\sin A = 3 \sin \frac{A}{3} - 4 \sin^{3} \frac{A}{3}$ , prove that we should expect to obtain also the value of $\sin \frac{π - A}{3}$ and $- \sin \frac{π + A}{3}$ .

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 33

If $\cos \frac{A}{3}$ be determined from the equation $\cos A = 4 \cos^{3} \frac{A}{3} - 3 \cos \frac{A}{3}$ , prove that we should expect to obtain also the value of $\cos \frac{2 π - A}{3}$ and $\cos \frac{2 π + A}{3}$ .

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