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Chapter 5: Problem 103

Convert the degree measure \(225^{\circ}\) to radians.

Short Answer

Expert verified
\( \frac{5}{4} \pi \)

Step by step solution

01

Identify the Formula

To convert degrees to radians, use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
02

Substitute the Degree Value

Substitute the degree value (225 degrees) into the formula:\[ 225^{\circ} \times \frac{\pi}{180} \]
03

Simplify the Expression

Simplify the fraction inside the formula:\[ 225 \times \frac{\pi}{180} = \frac{225}{180} \pi \]
04

Reduce the Fraction

Reduce the fraction \( \frac{225}{180} \) by dividing both numerator and denominator by their greatest common divisor, which is 45:\[ \frac{225 \div 45}{180 \div 45} \pi = \frac{5}{4} \pi \]
05

Express the Final Answer

The angle in radians is:\[ \frac{5}{4} \pi \]

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Key Concepts

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

Angle Conversion from Degrees to Radians
Angle conversion is the process of changing an angle's measurement from one unit to another. In this case, we are converting an angle from degrees to radians.
Here's the basic formula for converting degrees to radians:
\[\text{radians} = \text{degrees} \times \frac{\text{蟺}}{180}\]
In simpler terms, to convert an angle from degrees to radians, you multiply the number of degrees by \[ \frac{\text{蟺}}{180}.\]
This formula is crucial because it lets us shift between these two units easily.
Remember, \(\text{蟺}\) (pi) is approximately 3.14159. But, in most calculations, you can leave it as \(\text{蟺}\).
Understanding Radian Measure
When we talk about angles, radians are another way to measure them, similar to degrees but based on the properties of circles. A radian is the angle formed when the radius of a circle is wrapped along its circumference.
A full circle has \[2\text{蟺}\] radians which corresponds to 360 degrees. Consequently, a semicircle, or half circle, has \[ \text{蟺}\] radians, equating to 180 degrees.
Using radians can make some math more intuitive, especially in calculus and trigonometry.
To summarize, radians offer a natural way of measuring angles based on the radius and arc length of circles.
Simplifying Fractions in Angle Conversion
When converting degrees to radians, you often encounter a fraction you need to simplify. Simplifying fractions means reducing them to their smallest form.
For example, during the conversion process, we had \(\frac{225}{180} \). To simplify this, find the greatest common divisor (GCD) of 225 and 180. The GCD is 45 in this case.
We then divide both the numerator (225) and the denominator (180) by 45, which simplifies our fraction:
\[ \frac{225 \div 45}{180 \div 45} = \frac{5}{4} \]
This step is essential to make the final radian measure clearer and simpler to use.

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Most popular questions from this chapter

The graph of \(y=\cos (x)\) is shifted \(\pi\) units to the left, reflected in the \(x\) -axis, and then shifted 2 units upward. What is the equation of the curve in its final position?

Use a calculator to find the value of each function. Round answers to four decimal places. $$\sin \left(344.1^{\circ}\right)$$

Convert each angle to decimal degrees. When necessary, round to four decimal places. $$-5^{\circ} 45^{\prime} 30^{\prime \prime}$$

Determine the amplitude and phase shift for each function, and sketch at least one cycle of the graph. Label five points as done in the examples. $$y=-\sin (x)-1$$

Find the exact value of each of the following expressions without using a calculator. $$\tan \left(135^{\circ}\right)$$
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