Problem 21 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$170^{\circ}$$

Step-by-Step Solution

Answer

The angle \(170^{\circ}\) is \(\frac{17\pi}{18}\) in radians.

Step 1: Understand the degree to radian conversion formula

To convert an angle from degrees to radians, we use the conversion formula. The formula is: \( \text{radians} = \text{degrees} \times \left( \frac{\pi}{180} \right) \). This formula comes from the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 2: Substitute the given angle into the formula

Now we will substitute \(170^{\circ}\) into the conversion formula: \( 170 \times \left( \frac{\pi}{180} \right) \).

Step 3: Simplify the expression

Multiply the degree measure by \(\frac{\pi}{180}\): \( 170 \times \frac{\pi}{180} = \frac{170\pi}{180} \).

Step 4: Simplify the fraction

To simplify \(\frac{170\pi}{180}\), find the greatest common divisor of 170 and 180, which is 10. Dividing both the numerator and the denominator by 10 gives \(\frac{17\pi}{18}\).

Key Concepts

Degrees

Degrees are a unit of measurement for angles. They are commonly used in everyday situations, such as measuring turn angles or checking the angles in geometry. The circle is divided into 360 equal parts, hence we say a full circle has 360 degrees.

• Each degree represents \(\frac{1}{360}\) of a full circle.
• It is often denoted by the symbol "\(\degree\)".
• Angles in degrees make it easier for people to deal with fractions and percentages, since 360 is divisible by so many different numbers.

When dealing with trigonometry, using degrees can make raw calculations seem more intuitive, especially when starting out with basic angle measurements. However, in advanced mathematics, converting degrees to other units like radians is very important.

Radians

Radians provide another way to measure angles and are often used in various branches of mathematics, especially trigonometry. Unlike degrees, radians are based on the radius of a circle.

• One radian is the angle created when the arc length of a circle is equal to the radius of the circle.
• A full circle in radian measure is \(2\pi\), because the circumference of a circle is \(2\pi\) times its radius.
• Radians are especially useful in calculus and for simplifying trigonometric functions and identities.

Angles in radians can sometimes seem less intuitive at first, but they provide a more natural fit for mathematical formulas involving circular motion and periodic functions. When dealing with trigonometric functions like sine or cosine, radians make calculations simpler and more precise.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships involving lengths and angles of triangles. It's a key area that makes use of both degrees and radians for measurement. Understanding trigonometry is crucial for many fields, ranging from engineering to physics.

• Trigonometry functions (such as sine, cosine, and tangent) are heavily used to describe ratios of sides in right triangles.
• These functions have periodic properties, and understanding angle measurements in radians can be quite useful here.
• Topics like the unit circle are fundamental concepts that marry both radians and trigonometry, offering a geometric perspective on angle measurements.

When studying trigonometry, transitioning between degrees and radians becomes essential as certain problems might prefer one measurement over the other for simplicity and precision.

Conversion Formula

The conversion between degrees and radians in trigonometry is a simple yet crucial process that is encapsulated by a fundamental formula. The transition from degrees to radians is achieved using:\[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \]This formula stems from the key relationship that \(180\degree = \pi\) radians. By using this conversion formula, we can easily change angle measures from one system to another.

• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).
• This formula is derived from the circumference (or a complete revolution) of a circle being equal to \(2\pi\) radians or \(360\degree\).
• Using this conversion allows mathematical operations to be performed consistently in contexts where radians provide a more analytical framework.

Understanding this formula not only enhances comprehension of trigonometric functions but also integrates well with higher-level mathematics, where radian measure often becomes necessary.