Problem 14 - Algebra 2 and Trigonometry

Question

In \(13-22\) , find the degree measure of each angle whose radian measure is given. \(\frac{\pi}{9}\)

Step-by-Step Solution

Answer

The angle is 20 degrees.

Step 1: Convert Radians to Degrees

To convert radians to degrees, use the conversion factor \( \frac{180^\circ}{\pi} \). Multiply the radian measure by this factor.

Step 2: Substitute the Given Radian Value

Substitute \( \frac{\pi}{9} \) into the conversion formula: \( \frac{\pi}{9} \times \frac{180^\circ}{\pi} \).

Step 3: Cancel Common Terms

Cancel out \( \pi \) in the numerator and the denominator: \( \frac{1}{9} \times 180^\circ \).

Step 4: Perform the Multiplication

Calculate the remaining expression: \( \frac{180^\circ}{9} \).

Step 5: Simplify the Result

Simplify \( \frac{180^\circ}{9} \) by performing the division: \( 20^\circ \).

Key Concepts

Radian to Degree Conversion

Radians and degrees are two ways to measure angles. Converting between them is important, especially in math and science. To convert from radians to degrees, we use a conversion factor derived from the fact that a full circle is both 360 degrees and \(2\pi\) radians. Therefore, \(1\) radian equals \(\frac{180^\circ}{\pi}\).

• Start by identifying the radian measure given.
• Multiply this value by the conversion factor \(\frac{180^\circ}{\pi}\) to find the angle in degrees.
• It's important to simplify fractions and cancel out any common terms, such as \(\pi\), to make calculations easier.

For example, if you have an angle of \(\frac{\pi}{9}\) radians, you substitute this value into the formula and simplify to ultimately find the angle in degrees.

Angle Measurement

Angles are measured in various units, and understanding these units is key in trigonometry. While degrees are often used in everyday contexts, radians are more common in mathematical analysis and calculus.

• Degrees are based on dividing a circle into 360 parts. This makes it easy to visualize and measure angles such as \(90^\circ\) or \(45^\circ\).
• Radians, on the other hand, measure how many times the radius fits around the arc of the circle. This unit is more intuitive when dealing with circles and periodic functions.

Both units are essential, and you need to be able to switch between them easily to solve various mathematical problems. Knowing how many radians are in a degree and vice versa is crucial for calculations.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between angles and lengths in triangles. It has applications in various fields, from engineering to physics.

• The three basic trigonometric functions are sine, cosine, and tangent, each defining specific angle-to-side relations in right triangles.
• By understanding the concepts of radians and degrees, you gain valuable tools for calculating these functions and applying them to solve real-world problems.
• For example, if you know the angle in degrees, you can easily convert it to radians to use in trigonometric functions, which often require radian measures.

Trigonometry transcends simple triangle measurements, impacting wave patterns, circular movements, and even atomic physics. Thus, mastering angle conversions and measurements can significantly enhance your problem-solving capabilities in these areas.