Chapter 7: Q. 87 (page 487)

Write the trigonometric expression as an algebraic expression containing u and v
$\sin (\tan^{- 1} u - \sin^{- 1} v)$

Short Answer

Expert verified

$\sin (\tan^{- 1} u - \sin^{- 1} v) = \frac{u \sqrt{1 - v^{2}} - v}{\sqrt{u^{2} + 1}}$

Step by step solution

Step 1. Given information we have is

a trigonometric equation $\sin (\tan^{- 1} u - \sin^{- 1} v)$

Step 2. putting tan-1u=xu=tanxsin-1v=yv=siny

$u = \frac{\sin u}{\cos u}$

$u \cos u = \sin u$

Step 3. Using the Pythagoras theoremsin2x+cos2x=1

${(u \cos x)}^{2} + \cos^{2} x = 1$

$u^{2} \cos^{2} x + \cos^{2} x = 1 \cos^{2} x (u^{2} + 1) = 1 \cos^{2} x = \frac{1}{u^{2} + 1} \cos x = \frac{1}{\sqrt{u^{2} + 1}}$

Now,

$\sin x = \frac{u}{\sqrt{u^{2} + 1}}$

Step 4. Now, we know thatcosy=1-sin2ycosy=1-v2using the formula ofsinx-y=sinxcosy-cosxsiny we get

On Solving

= \frac{u}{\sqrt{u^{2} + 1}} \sqrt{1 - v^{2}} - \frac{v}{\sqrt{u^{2} + 1}} = \frac{u \sqrt{1 - v^{2}} - v}{\sqrt{u^{2} + 1}}

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Write the trigonometric expression as an algebraic expression containing u and v
$\sin (\tan^{- 1} u - \sin^{- 1} v)$

Short Answer

Step by step solution

Step 1. Given information we have is

Step 2. putting tan-1u=xu=tanxsin-1v=yv=siny

Step 3. Using the Pythagoras theoremsin2x+cos2x=1

Step 4. Now, we know thatcosy=1-sin2ycosy=1-v2using the formula ofsinx-y=sinxcosy-cosxsiny we get

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Write the trigonometric expression as an algebraic expression containing u and vsintan-1u-sin-1v

Short Answer

Step by step solution

Step 1. Given information we have is

Step 2. putting tan-1u=xu=tanxsin-1v=yv=siny

Step 3. Using the Pythagoras theoremsin2x+cos2x=1

Step 4. Now, we know thatcosy=1-sin2ycosy=1-v2using the formula ofsinx-y=sinxcosy-cosxsiny we get

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Calculus

Probability and Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Write the trigonometric expression as an algebraic expression containing u and v
$\sin (\tan^{- 1} u - \sin^{- 1} v)$