Question

Convert each angle in radians to degrees. \(-\frac{13 \pi}{6}\)

Short answer

-390 degrees

Step-by-step solution

Understand the Conversion Formula

The formula to convert an angle from radians to degrees is: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\text{π}} \]

Apply the Conversion Formula

Given the angle \[ -\frac{13 \text{π}}{6} \], substitute it into the conversion formula:\[ \text{Degrees} = -\frac{13 \text{π}}{6} \times \frac{180}{\text{π}} \]

Simplify the Expression

Simplify the expression by canceling out \(\text{π}\) and performing the multiplication:\[ \text{Degrees} = -\frac{13}{6} \times 180 \] \[ \text{Degrees} = -13 \times 30 \] \[ \text{Degrees} = -390 \]

Key concepts

radian to degree conversion

Radians and degrees are two units for measuring angles. In many mathematical problems, especially in trigonometry, you may need to convert angles from radians to degrees.
The formula to convert radians to degrees is: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\text{π}} \]
This formula works because a full circle is both 360 degrees and \(2\text{π}\) radians. Therefore, 1 radian is equal to \(\frac{180}{\text{π}}\) degrees.
Let’s go through an example: Convert \(-\frac{13 \text{π}}{6}\) radians to degrees.
Step-by-step, the conversion looks like this: \[ \text{Degrees} = -\frac{13 \text{π}}{6} \times \frac{180}{\text{π}} \]

First, notice how the \(\text{π}\) terms cancel out, simplifying to: \[ \text{Degrees} = -\frac{13}{6} \times 180 \]
Then, perform the multiplication: \[ \text{Degrees} = -13 \times 30 \] \[ \text{Degrees} = -390 \]
Thus, \(-\frac{13 \text{π}}{6}\) radians equals -390 degrees.

angle measurement

Angles are a fundamental concept in geometry and trigonometry. They measure the rotation between two intersecting lines. We often measure angles in either degrees or radians.

Degrees measure parts of a full circle, where a full circle is 360 degrees. For example:

• 90 degrees represent a right angle.
• 180 degrees represent a straight line.

Conversely, radians often appear in higher mathematics and physics. Radians measure angles based on the radius of a circle. One radian is the angle formed when the arc's length equals the radius length, making a full circle equal to \(2\text{π}\) radians.
Understanding the relationship between degrees and radians helps you convert between these units accurately. Always use the formula:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\text{π}} \]. This formula makes switching between radians and degrees straightforward.

trigonometry

Trigonometry is the branch of mathematics that studies angles and their relationships. It often involves converting between radians and degrees. Understanding trigonometry is crucial for solving problems related to triangles, waves, and oscillations.

Key functions in trigonometry include sine, cosine, and tangent. These functions relate the angles of a triangle to the lengths of its sides. For example:

• The sine function \(\text{sin} (θ)\) gives the ratio of the opposite side to the hypotenuse.
• The cosine function \(\text{cos} (θ)\) gives the ratio of the adjacent side to the hypotenuse.
• The tangent function \(\text{tan} (θ)\) gives the ratio of the opposite side to the adjacent side.

Trig functions use radians often, so knowing how to convert between radians and degrees is useful. The conversion formula \[ \text{Radians} = \text{Degrees} \times \frac{\text{π}}{180} \] lets you easily switch from degrees to radians. Overall, mastering radians, degrees, and trigonometric functions is vital for anyone studying higher mathematics or physics.