Question

Convert each angle in radians to degrees. \(\frac{5 \pi}{12}\)

Short answer

75 degrees

Step-by-step solution

Understand the Conversion Factor

To convert an angle from radians to degrees, use the conversion factor \(180^\circ = \pi \text{ radians}\).

Set Up the Conversion

Multiply the given radian measure by the conversion factor \(\frac{180^\circ}{\pi \text{ radians}}\). For \( \frac{5\pi}{12} \): \( \frac{5\pi}{12} \times \frac{180^\circ}{\pi} \).

Simplify the Expression

Cancel out \( \pi \) in the numerator and denominator: \(( \frac{5\pi}{12} \times \frac{180^\circ}{\pi}) = \frac{5 \times 180^\circ}{12}\).

Calculate the Degrees

Perform the multiplication and division: \( \frac{5 \times 180}{12} = \frac{900}{12} = 75^\circ \).

Key concepts

radians to degrees

Converting radians to degrees is a vital skill in trigonometry and calculus. The conversion process relies on the relationship between radians and degrees. Specifically, one complete revolution around a circle is equal to \(360^\text{°}\) or \(2\text{π}\) radians. Thus, \(180^\text{°}\) is equivalent to \(π\) radians.

To convert from radians to degrees, you can use the following conversion factor:

\[1 \text{ radian} = \frac{180^\text{°}}{\text{π}}\text{ radians}\]

By multiplying the radian measure by this factor, you can easily convert it to degrees. For example, to convert \(\frac{5\text{π}}{12}\) radians to degrees, as shown in the solution:

\(\frac{5\text{π}}{12} \times \frac{180^\text{°}}{\text{π}} = \frac{5 \times 180^\text{°}}{12}\)

When you carry out the multiplication and division, you get:

\(\frac{900}{12} = 75^\text{°}\)

Now you know that \(\frac{5\text{π}}{12}\) radians is equal to \(75^\text{°}\).

angle measurement

Understanding angle measurement is crucial in geometry and trigonometry. Angles can be measured in degrees or radians, which are two scales used to describe the same concept.

Degrees are the most common unit. A full circle is divided into 360 degrees, with each degree further divided into 60 minutes, and each minute into 60 seconds.

Radians, however, are based on the radius of a circle. One radian is the angle created when the arc length of a circle is equal to its radius. There are \(2π\) radians in a full circle.

Here are the key points to understand about angle measurement:

• Degrees: 1 complete revolution = 360°.
• Radians: 1 complete revolution = 2π radians.
• Relationship: 180° = π radians, hence 1° = π/180 radians and 1 radian = 180/π degrees.

Understanding these relationships is crucial for converting between radians and degrees and for solving various trigonometric problems.

trigonometry basics

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles. It is foundational for many fields, from engineering to physics.

Here are a few basic concepts:

• Triangles and Pythagorean Theorem: In a right-angled triangle: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
• Trigonometric Ratios: Sine (sin), Cosine (cos), and Tangent (tan). These ratios relate the angles of a triangle to the lengths of its sides. For an angle θ: \(\text{sin} θ = \frac{\text{Opposite}}{\text{Hypotenuse}}\), \(\text{cos} θ = \frac{\text{Adjacent}}{\text{Hypotenuse}}\), \(\text{tan} θ = \frac{\text{Opposite}}{\text{Adjacent}}\).
• Unit Circle: A circle with radius 1, centered at the origin of the coordinate plane, used to define trigonometric functions for all angles.

These basics help in understanding more complex trigonometric concepts such as identities, equations, and their applications in real-world problems.