8-1P Express the following angles in ... [FREE SOLUTION]

Chapter 6: 8-1P (page 138)

Express the following angles in radians: (a) 45.0掳, (b) 60.0掳, (c) 90.0掳, (d) 360.0掳, and (e) 445掳. Give as numerical values and as fractions of \(\pi \).

Short Answer

Expert verified

The radian values of the given angles are (a) \(\frac{\pi }{4}\;{\rm{rad}}\), (b) \(\frac{\pi }{3}\;{\rm{rad}}\), (c) \(\frac{\pi }{2}\;{\rm{rad}}\), (d) \(2\pi \;{\rm{rad}}\), and \(\frac{{89\pi }}{{36}}\;{\rm{rad}}\).

Step by step solution

Process to change the unit from degree to radian

To change the angle from degree to radian, you should multiply it by \(\frac{\pi }{{{{180}^ \circ }}}\).

Calculation

Part (a)

Now, expressing \({45.0^ \circ }\) in radians unit, you get:

\(\begin{aligned}{45.0^ \circ } &= {45.0^ \circ } \times \frac{\pi }{{{{180}^ \circ }}}\;{\rm{rad}}\\ &= \frac{\pi }{4}\;{\rm{rad}}\end{aligned}\)

Part (b)

Express \({60^ \circ }\) in radians unit.

\(\begin{aligned}{60^ \circ } &= {60^ \circ } \times \frac{\pi }{{{{180}^ \circ }}}\;{\rm{rad}}\\ &= \frac{\pi }{3}\;{\rm{rad}}\end{aligned}\)

Part (c)

Now, expressing \({90^ \circ }\) in radians unit, you get:

\(\begin{aligned}{90^ \circ } &= {90^ \circ } \times \frac{\pi }{{{{180}^ \circ }}}\;{\rm{rad}}\\ &= \frac{\pi }{2}\;{\rm{rad}}\end{aligned}\)

Part (d)

Express \({360^ \circ }\) in radians unit.

\(\begin{aligned}{360^ \circ } &= {360^ \circ } \times \frac{\pi }{{{{180}^ \circ }}}\;{\rm{rad}}\\ &= 2\pi \;{\rm{rad}}\end{aligned}\)

Part (e)

Now, expressing \({360^ \circ }\) in radians unit, you get:

\(\begin{aligned}{445^ \circ } &= {445^ \circ } \times \frac{\pi }{{{{180}^ \circ }}}\;{\rm{rad}}\\ &= \frac{{89\pi }}{{36}}\;{\rm{rad}}\end{aligned}\)