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Chapter 4: Problem 79

Determine whether the statement is true or false. Justify your answer. $$\sin 60^{\circ} \csc 60^{\circ}=1$$

Short Answer

Expert verified
The statement is true.

Step by step solution

01

Evaluating Sine

First, evaluate \(\sin(60^{\circ})\). According to the standard values in trigonometry, \(\sin(60^{\circ}) = \sqrt{3}/2\) which is not equal to zero.
02

Evaluating the Expression

Now substitute \(\sin(60^{\circ})\) and \(\csc(60^{\circ})\) in the given expression. Hence, \(\sin(60^{\circ}) \csc(60^{\circ})\) becomes \((\sqrt{3}/2) * (2/\sqrt{3})\). This results in \(\sqrt{3}/2 * 2/\sqrt{3} = (\sqrt{3} * 2) / (2 * \sqrt{3})\). Since the numerator and the denominator are equal, the fraction simplifies to 1.
03

Conclusion

Therefore, the given statement \(\sin(60^{\circ}) \csc(60^{\circ}) = 1\) is true, as the left hand side simplifies to the right hand side. Justification: The number 1 divided or multiplied by itself always equals 1.

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