Problem 55
Question
Convert from degrees to radians. Round your answers to three significant digits. $$56.5^{\circ}$$
Step-by-Step Solution
Verified Answer
The angle in radians is approximately 3.09.
1Step 1: Understanding the Conversion Formula
To convert an angle from degrees to radians, we use the formula \( ext{Radians} = ext{Degrees} imes rac{oldsymbol{oldsymbol{ ext{π}}}}{180} \). This formula is derived from the fact that a full circle is \(360\) degrees or \(2\pi\) radians.
2Step 2: Applying the Conversion Formula
Substitute \(56.5^{ ext{o}}\) into the formula: \( ext{Radians} = 56.5 imes rac{oldsymbol{oldsymbol{ ext{π}}}}{180} \).
3Step 3: Calculating the Radian Measure
Perform the multiplication: \( 56.5 imes rac{oldsymbol{oldsymbol{ ext{π}}}}{180} \approx 0.985 \pi \).
4Step 4: Rounding to Three Significant Digits
Convert \(0.985 \pi\) to a decimal approximation using \(\pi \approx 3.14159\), resulting in \(0.985 \times 3.14159 \approx 3.09\). Finally, round \(3.09\) to three significant digits, which is already done here.
Key Concepts
TrigonometryAngular MeasurementRadian
Trigonometry
Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It plays a crucial role in many fields such as physics, engineering, and computer science. At its most basic, trigonometry helps us understand geometric properties and solve problems involving triangles. Since triangles form the basis of many geometric figures, this knowledge is pivotal.
Some key terms in trigonometry include sine, cosine, and tangent. These are functions that relate the angles of a right triangle to the ratios of its sides. For instance, the sine of an angle is the ratio of the opposite side to the hypotenuse. Similarly, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
Trigonometry isn't limited to triangles with right angles; it extends to understand circular motion and waves. Hence the need to understand different angle measurements like degrees and radians becomes important. As we advance, we'll see that radians are especially useful in calculus and higher-level math.
Some key terms in trigonometry include sine, cosine, and tangent. These are functions that relate the angles of a right triangle to the ratios of its sides. For instance, the sine of an angle is the ratio of the opposite side to the hypotenuse. Similarly, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
Trigonometry isn't limited to triangles with right angles; it extends to understand circular motion and waves. Hence the need to understand different angle measurements like degrees and radians becomes important. As we advance, we'll see that radians are especially useful in calculus and higher-level math.
Angular Measurement
Angular measurement is a way of describing the size of an angle. The most commonly used units for measuring angles are degrees and radians. Degrees are more familiar to most people; a full circle divides into 360 degrees.
However, mathematicians often prefer radians for more complex calculations. Radians provide a natural way to describe angles, especially when working with trigonometric functions.
However, mathematicians often prefer radians for more complex calculations. Radians provide a natural way to describe angles, especially when working with trigonometric functions.
- Degrees: A degree is a measure of angle equal to 1/360 of a full circle. It is denoted by the symbol \(^\circ\).
- Radians: A radian is based on the radius of a circle. One radian equals the angle formed when the length of the arc of the circle is equal to the circle's radius. There are \(2\pi\) radians in a full circle.
Radian
A radian is another unit to measure angles, and it's derived from the properties of a circle. One radian equals the angle created when you take the radius of a circle and lay it along the circle's edge. This is a natural way to measure angles because it directly relates to the properties of the circle itself.
In more technical terms, since the circumference of a circle is \(2\pi\) times the radius, there are \(2\pi\) radians in a full circle. Thus:
In more technical terms, since the circumference of a circle is \(2\pi\) times the radius, there are \(2\pi\) radians in a full circle. Thus:
- 180 degrees is equivalent to \pi\ radians.
- 90 degrees, which is a right angle, is \frac{\pi}{2}\ radians.
- Understanding radians is incredibly important in higher mathematics because they simplify many formulas and calculations. For example, the derivative of trigonometric functions like sine and cosine in radians are far simpler than in degrees.
When converting degrees to radians, you utilize the straightforward formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]. By mastering this conversion, you lay a strong foundation for tackling more advanced mathematical problems.
Other exercises in this chapter
Problem 55
A 40-foot slide leaning against the bottom of a building's window makes a \(55^{\circ}\) angle with the building. The angle formed with the building by the line
View solution Problem 55
Determine whether each statement is possible or not. $$\cot \theta=500$$
View solution Problem 56
Determine whether each statement is possible or not. $$\sec \theta=0.9996$$
View solution Problem 56
Convert from degrees to radians. Round your answers to three significant digits. $$298.7^{\circ}$$
View solution