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Chapter 7: Problem 2

Find the exact radian measure of the angle given in degree measure. $$150^{\circ}$$

Short Answer

Expert verified
The exact radian measure of \(150^{\circ}\) is \(\frac{5}{6}\pi\).

Step by step solution

01

Understand Radians and Degrees

Degrees and radians are two units used to measure angles. One full circle is equal to 360 degrees or \(2\pi\) radians. We need to convert from degrees to radians in this problem.
02

Use the Conversion Formula

To convert degrees to radians, use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \].We will apply this formula to convert \(150^{\circ}\).
03

Substitute the Given Value

Substitute \(150\) degrees into the conversion formula: \[ 150 \times \frac{\pi}{180} \].
04

Simplify the Fraction

Simplify the fraction \(\frac{150}{180}\). The greatest common divisor of 150 and 180 is 30, so divide the numerator and the denominator by 30:\[ \frac{150}{180} = \frac{5}{6} \].
05

Calculate the Radian Measure

Multiply the simplified fraction by \(\pi\):\[ \frac{5}{6} \pi \].So, \(150^{\circ}\) converted to radians is \(\frac{5}{6}\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees to Radians
When dealing with angles, it鈥檚 essential to understand the units used. Degrees and radians are the two main units for measuring angles. While degrees are more common in everyday uses, radians are very important in mathematics, especially in calculus and trigonometry.
  • One full circle equals 360 degrees.
  • In radian measure, a full circle is denoted as \(2\pi\) radians.
  • This means 180 degrees is the equivalent of \(\pi\) radians.
To convert any degree measure to radians, you can use the formula:\[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]This conversion factor \(\frac{\pi}{180}\) stems from the proportion that 180 degrees correspond to \(\pi\) radians.
Angle Conversion
Angle conversion is the process of changing an angle from one unit of measure to another. In the case of degrees to radians conversion, we apply the formula provided above. Consider the example from the original problem: converting \(150^{\circ}\) to radians.
Firstly, substitute the given angle into the conversion formula: \[150 \times \frac{\pi}{180}\]
By multiplying, you apply the conversion factor \(\frac{\pi}{180}\) to the degree measure 150. This effectively translates the angle into its equivalent radian measure. It's a straightforward calculation, but understanding why the conversion factor is used is key to becoming comfortable with angle conversions.
Mathematical Simplification
In conversion problems, simplifying terms can make calculations easier and more elegant. When we substitute \(150^{\circ}\) into the conversion formula, we get \[\frac{150}{180} \times \pi\]
Simplifying \(\frac{150}{180}\) helps us express the radian measure in reduced form. The greatest common divisor of 150 and 180 is 30, so divide both the numerator and the denominator by 30: \[\frac{150}{180} = \frac{5}{6}\]
This results in a simpler fraction: \[\frac{5}{6}\pi\]Thus, the radian measure for \(150^{\circ}\) effectively becomes \(\frac{5}{6}\pi\), showing the importance of mathematical simplification in making problems easier to solve and understand.

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Most popular questions from this chapter

Solve each equation for \(x\) in the given interval. Give answers exactly, if possible. Otherwise, give answers accurate to three significant figures. $$2 \cos x+\sin 2 x=0,-180^{\circ}

Solve the equation for the stated solution interval. Find exact solutions when possible, otherwise give solutions to three significant figures. Verify solutions with your GDC. $$2 \cos ^{2} x+\cos x=0,0 \leqslant x<2 \pi$$

Solve each equation for \(x\) in the given interval. Give answers exactly, if possible. Otherwise, give answers accurate to three significant figures. Find an identity for \(\sin 3 x\) in terms of \(\sin x\)

$$\text { Show that } \arctan \frac{1}{2}+\arctan \frac{1}{3}=\frac{\pi}{4}$$.

Prove each identity. $$\frac{\cos 2 \theta}{\cos \theta+\sin \theta}=\cos \theta-\sin \theta$$
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