PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
8th Edition
ISBN: 9781337904285
Author: Larson
Publisher: CENGAGE L
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Concept explainers

Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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Chapter 5.4, Problem 53E
To determine

To find: The algebraic form of the given trigonometric expression.

Expert Solution & Answer
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Answer to Problem 53E

The algebraic form of the trigonometric expression sin(arctan2x−arccosx) is 2x2−1−x21+4x2 .

Explanation of Solution

Given information:

The trigonometric expression sin(arctan2x−arccosx) .

Formula used:

Difference rule for sine function is sin(x−y)=sinxcosy−sinycosx .

Calculation:

Consider trigonometric expression sin(arctan2x−arccosx) .

The above expression is of the form sin(u−v) , where u=arctan2x and v=arccosx .

Recall difference rule for sine function is sin(x−y)=sinxcosy−sinycosx .

Trigonometric and its inverse trigonometric function cancel each other.

Apply it,

  sin(arctan2x−arccosx)=sin(arctan2x)cos(arccosx)−cos(arctan2x)sin(arccosx)=sin(arctan2x)⋅x−cos(arctan2x)sin(arccosx)

For u=arctan2x .

Recall that tanx=perpendicularbase

Construct a right angle triangle with perpendicular p as 2x and base b as 1.

Since, h2=p2+b2 , where h is hypotenuse of triangle.

So, hypotenuse h of triangle is 1+4x2 .

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.4, Problem 53E , additional homework tip 1

For v=arccosx .

Recall that cosx=basehypotenuse

Construct a right angle triangle with base b as x and hypotenuse h as 1.

Since, h2=p2+b2 , where p is perpendicular of triangle.

So, perpendicular p of triangle is 1−x2 .

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.4, Problem 53E , additional homework tip 2

The expression is reduced to,

  sin(arctan2x−arccosx)=sin(arcsin2x1+4x2)⋅x−cos(arccos11+4x2)sin(arcsin1−x21)=2x21+4x2−1−x21+4x2=2x2−1−x21+4x2

Thus, the algebraic form of the trigonometric expression sin(arctan2x−arccosx) is 2x2−1−x21+4x2 .

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Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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