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Chapter 4: Problem 80

Determine whether the statement is true or false. Justify your answer. $$\sec 30^{\circ}=\csc 60^{\circ}$$

Short Answer

Expert verified
The statement is true, both \( \sec 30^{\circ} \) and \( \csc 60^{\circ} \) are equal to \( \frac{2\sqrt{3}}{3} \).

Step by step solution

01

Calculation of sec 30掳

First, calculate the value of sec 30掳. The secant of an angle is the reciprocal of the cosine of the angle. So, \( \sec 30^{\circ} = \frac{1}{\cos 30^{\circ}}\). The cosine of 30掳 is \(\frac{\sqrt{3}}{2}\), therefore, \( \sec 30^{\circ} = \frac{2}{\sqrt{3}} \) which simplifies to \( \frac{2\sqrt{3}}{3} \).
02

Calculation of csc 60掳

Next, calculate the value of csc 60掳. The cosecant of an angle is the reciprocal of the sine of the angle. So, \( \csc 60^{\circ} = \frac{1}{\sin 60^{\circ}}\). The sine of 60掳 is \(\frac{\sqrt{3}}{2}\), therefore, \( \csc 60^{\circ} = \frac{2}{\sqrt{3}} \) which simplifies to \( \frac{2\sqrt{3}}{3} \).
03

Comparison

Now that both sides of the equation are calculated, compare the results. Both sec 30掳 and csc 60掳 are equal to \( \frac{2\sqrt{3}}{3} \).

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