Problem 30 - Algebra and Trigonometry

Question

In Exercises \(29-34,\) convert each angle in degrees to radians. Round to two decimal places. $$ 76^{\circ} $$

Step-by-Step Solution

Answer

The angle of 76 degrees is approximately 1.33 radians.

Step 1: Understanding the Conversion Formula

When converting from degrees to radians, the formula to use is radians = degrees * \(\pi/180\). This formula is derived from the fact that the circle has 360 degrees or \(2\pi\) radians, hence, 1 degree equals \(2\pi/360\) or \(\pi/180\) radians.

Step 2: Apply the Conversion Formula

For this problem, the angle given is 76 degrees. So, using the formula, the radians will be \(76 * \(\pi/180\) = 1.33 radians. Make sure to round the result up to two decimal positions as requested.

Key Concepts

Angle Measurement

Angles are measured using two common units: degrees and radians. Degrees are what many of us are familiar with from geometry classes. A full circle is 360 degrees. This means if you start at one point on the circle and go all the way around back to the start, you’ve covered 360 degrees.
Radians, on the other hand, are a less intuitive unit for many, but they are very important in trigonometry and calculus due to their natural properties in mathematical formulas. A complete circle in radians is represented by the value of \(2\pi\).
Understanding the foundational units of angle measurement is critical when working with angles in various mathematical fields.

• Degree: A circle is divided into 360 equal parts called degrees.
• Radian: A circle is divided into \(2\pi\) equal parts, providing a smooth mathematical transition between linear and circular dimensions.

Recognizing that both are simply different ways of measuring the same thing helps when transitioning between units.

Trigonometric Conversion

Trigonometric conversion involves changing an angle from one unit of measurement to another. When dealing with trigonometric functions, often angles need to be in radians for functions like sine, cosine, and tangent.
The conversion process from degrees to radians is straightforward. Use the conversion formula:

• Radians = Degrees × \(\frac{\pi}{180}\)

For instance, to convert 76 degrees into radians, we calculate:\[76 \times \frac{\pi}{180} = 1.33\] radians (rounded to two decimal places).
This formula comes from the relationship between the circumference of a circle (\(2\pi\)) and its division into 360 degrees.

• This ensures that any angle can be smoothly converted, preserving the trigonometric properties.
• This conversion is essential in higher mathematics for integrating and differentiating functions involving angles.

Radian Calculation

When you calculate radians, you are converting a degree measure into its corresponding angle on a circular arc. A direct connection exists between the arc's length and the radius of the circle, called the radian measure.
The radian is defined based on the radius of a circle:

• If you wrap the circle's radius along its circumference, the angle subtended at the center of the circle is 1 radian.

This unique property makes calculations involving radians much more straightforward and precise, especially in calculus. For our specific calculation of 76 degrees to radians, the operation path is:1. Multiply 76 degrees by \(\frac{\pi}{180}\).2. Solve the result to get the measure in radians.3. For this particular angle, the final radian measure is around 1.33 after rounding.
Remember:

• Radians are crucial for computing advanced mathematical problems.
• Converting measures helps align your analysis perfectly with trigonometric and calculus practices.