Question

Convert each angle in degrees to radians. Express your answer as a multiple of \(\pi .\) \(270^{\circ}\)

Short answer

\frac{3}{2} \pi

Step-by-step solution

Understand the conversion formula

To convert an angle from degrees to radians, use the conversion formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

Apply the conversion formula to 270 degrees

Substitute 270 degrees into the formula: \( \text{radians} = 270 \times \frac{\pi}{180} \)

Simplify the fraction

Simplify the fraction: \( 270 \times \frac{1}{180} = \frac{270}{180} = \frac{3}{2} \)

Combine the simplified fraction with \pi

Multiply the simplified fraction by \pi: \( \frac{3}{2} \pi \)

Final Answer

Therefore, 270 degrees is equal to \( \frac{3}{2} \pi \) radians when expressed as a multiple of \pi.

Key concepts

angle conversion

Angle conversion is the process of changing the measurement of an angle from one unit to another. One of the most common conversions is between degrees and radians. Degrees and radians are just two different ways of measuring angles.
Degrees are typically used in everyday contexts, like telling time or measuring angles in geometry. One full circle is 360 degrees.
Radians, on the other hand, are often used in higher mathematics, such as calculus and trigonometry. One full circle is 2\pi radians.
The key formula to convert between these units is:

• \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
  For example, if you have an angle of 270 degrees and want to convert it to radians, you would multiply by \(\frac{\pi}{180}\): \(270 \times \frac{\pi}{180} \).
  This simplifies to \( \frac{3}{2} \pi \).
  So, 270 degrees is \( \frac{3}{2} \pi \) radians. Converting angles is a crucial skill in trigonometry and various branches of mathematics.

radian measure

The radian measure is another way to express angles, often used in advanced mathematics.
A radian is defined based on the radius of a circle. Imagine wrapping the radius of the circle along its circumference; the angle covered by this arc length is one radian.
One important thing to remember is that there are 2\pi radians in a full circle, which corresponds to 360 degrees.
This relationship is very useful for conversions.
Consider these key points:

• Half a circle (or 180 degrees) is \( \pi \) radians.
• A right angle (or 90 degrees) is \( \frac{\pi}{2} \) radians.
• A 45-degree angle is \( \frac{\pi}{4} \) radians.

Because radians are based on the radius and the circumference of circles, they simplify many calculations in trigonometry, calculus, and physics.

trigonometry basics

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles.
It has many applications in science, engineering, and everyday life.
Here are some basic concepts:

• The primary functions in trigonometry are sine (sin), cosine (cos), and tangent (tan).
• These functions relate an angle of a right triangle to the ratios of two of its sides.

For example, for an angle \( \theta \):

• \( \sin(\theta) = \frac{opposite}{hypotenuse} \)
• \( \cos(\theta) = \frac{adjacent}{hypotenuse} \)
• \( \tan(\theta) = \frac{opposite}{adjacent} \)

Understanding these ratios is essential for solving problems involving triangles.
Finally, another important aspect is the unit circle, which helps to define these functions for all angles, not just those in right triangles.
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane.
Each point on the unit circle corresponds to an angle and can be expressed in terms of sine and cosine.
Mastering these basics will greatly aid in understanding more complex trigonometric concepts.