Which of the following is equivalent to [tex]\sin \theta \csc(-\theta)[/tex] wherever [tex]\sin \theta \csc(-\theta)[/tex] is defined?
F. [tex]-1[/tex]
G. [tex]1[/tex]
H. [tex]-\tan \theta[/tex]
J. [tex]\tan \theta[/tex]
K. [tex]-\sin^2 \theta[/tex]
Answer (3)
s in θ csc ( − θ ) = s in θ ⋅ s in ( − θ ) 1 = s in θ ⋅ − s in θ 1 = − 1 1 = − 1 A n s w er : F
The expression sin θ csc(–θ) simplifies to –1 by recognizing that csc(–θ) is –csc(θ). As a result, sin θ csc(–θ) = –1. Therefore, the correct answer is F. –1.
To determine which expression is equivalent to sin θ csc(–θ) wherever it is defined, let's break down the problem:
First, recall that csc(θ) is the reciprocal of sin(θ):
csc(θ) = 1/sin(θ)
Now, consider that the cosecant function is odd, meaning:
csc(–θ) = –csc(θ)
This gives us the following transformation:
sin(θ) csc(–θ) = sin(θ) imes (–csc(θ)) = sin(θ) imes (–1/sin(θ)) = –1
Thus, the expression simplifies to –1.
The correct answer is: F. –1
The expression sin θ csc ( − θ ) simplifies to − 1 . Therefore, the correct answer is option F. − 1 .
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