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Chapter 5: Problem 37

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\sin \phi(\csc \phi-\sin \phi)$$

Short Answer

Expert verified
The simplified form of the given expression \(\sin \phi(\csc \phi - \sin \phi)\) is \(\cos^2 \phi\).

Step by step solution

01

Identify the Fundamental identities

The fundamental identities to be used here are the reciprocal and Pythagorean identities. The reciprocal identity states that \(\csc \phi = 1/\sin \phi\).
02

Replace \(\csc \phi\) in the expression with the reciprocal identity

Substitute \(\csc \phi\) with \(1/\sin \phi\) in the given expression: \(\sin \phi (1/\sin \phi - \sin \phi)\).
03

Simplify the expression

The expression can now be simplified. The \(\sin \phi\) in the parenthesis cancels out with the \(\sin \phi\) outside the parenthesis for the first term. The expression then becomes: \(1 - \sin^2 \phi\).
04

Use the Pythagorean identity to get the final simplified form

The Pythagorean identity in trigonometry states \(1 = \sin^2 \phi + \cos^2 \phi\). In the expression \(1-\sin^2 \phi\), we can replace \(1\) with \(\sin^2 \phi + \cos^2 \phi\), yielding \(\sin^2 \phi + \cos^2 \phi - \sin^2 \phi\). This further simplifies to \(\cos^2 \phi\).

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Most popular questions from this chapter

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by $$y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t$$.where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (A). Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C)\). (B) Find the amplitude of the oscillations of the weight. (C) Find the frequency of the oscillations of the weight.

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Determine whether the statement is true or false. Justify your answer. $$\sin \left(x-\frac{\pi}{2}\right)=-\cos x$$

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