Problem 67
Question
Convert the angle measure from degrees to radians. Round to three decimal places. $$ -216.35^{\circ} $$
Step-by-Step Solution
Verified Answer
The measure of the angle in radians, rounded to three decimal places, can be obtained by applying the conversion formula and then rounding off to the appropriate decimal places.
1Step 1: Identify the given measure
The given angle measure is \(-216.35^{\circ}\).
2Step 2: Conversion formula
To convert degrees into radians, you use the following formula: radians = degrees × \(\pi /180\).
3Step 3: Substitute the value
Substitute \(-216.35^{\circ}\) into the formula: radians = \(-216.35 × \pi /180\).
4Step 4: Calculate
Calculate the result to obtain the angle measure in radians.
5Step 5: Round to 3 decimal places
Lastly, round off the result to three decimal places as indicated by the exercise.
Key Concepts
Degrees to RadiansRadian MeasureConversion Formula for Angles
Degrees to Radians
Converting angles from degrees to radians is a common task in trigonometry and calculus. Degrees and radians are just two different units for measuring angles, similar to how miles and kilometers measure distance. Understanding this conversion allows you to switch seamlessly between these units when necessary.
To convert an angle from degrees to radians, we use a specific mathematical approach. One important fact to remember is that a full circle is 360 degrees, which is equivalent to approximately 6.28319 radians (or in exact terms, it is \(2\pi\) radians). This gives us the key conversion factor: \(\pi\) radians equals 180 degrees. By establishing this relationship, you can easily convert any angle from one unit to the other.
This conversion is essential in many fields such as physics, engineering, and computer graphics, where angles need to be used in different mathematical contexts. By mastering this technique, you can enhance your problem-solving skills across these areas.
To convert an angle from degrees to radians, we use a specific mathematical approach. One important fact to remember is that a full circle is 360 degrees, which is equivalent to approximately 6.28319 radians (or in exact terms, it is \(2\pi\) radians). This gives us the key conversion factor: \(\pi\) radians equals 180 degrees. By establishing this relationship, you can easily convert any angle from one unit to the other.
This conversion is essential in many fields such as physics, engineering, and computer graphics, where angles need to be used in different mathematical contexts. By mastering this technique, you can enhance your problem-solving skills across these areas.
Radian Measure
Radian measure is another way to express angles, and it provides a natural way to relate angles to the geometry of a circle. Unlike degrees, which have an arbitrary scale based on historical reasons, radians relate directly to the size of the circle.
The concept of radians is based on the length of the arc that an angle subtends on a unit circle. In a circle with a radius of one, the length of the entire circumference is \(2\pi\). If you have an angle that measures 1 radian, the length of the arc it subtends is precisely that same measure of the angle.
The radian is particularly useful in calculus and more advanced mathematics because it simplifies the relationship between angles, arc lengths, and radii. Functions like sine, cosine, and tangent have their simplest and most natural expressions when angles are measured in radians.
The concept of radians is based on the length of the arc that an angle subtends on a unit circle. In a circle with a radius of one, the length of the entire circumference is \(2\pi\). If you have an angle that measures 1 radian, the length of the arc it subtends is precisely that same measure of the angle.
The radian is particularly useful in calculus and more advanced mathematics because it simplifies the relationship between angles, arc lengths, and radii. Functions like sine, cosine, and tangent have their simplest and most natural expressions when angles are measured in radians.
Conversion Formula for Angles
The conversion formula for angles is a simple yet powerful tool that translates angles between degrees and radians. The central formula is:
The formula enables you to find the radian equivalent of an angle provided in degrees, which is critical for dealing with math problems where angles need to be expressed in one standard unit. For example, when you need to convert \(-216.35^{\circ}\) into radians, simply plug the degree measure into the equation:
\(-216.35 \times \frac{\pi}{180}\).
This step-by-step conversion process not only enhances understanding but also ensures accuracy, especially when the exercise requires the result to be rounded to three decimal places.
- Radians = Degrees \(\times \frac{\pi}{180}\)
The formula enables you to find the radian equivalent of an angle provided in degrees, which is critical for dealing with math problems where angles need to be expressed in one standard unit. For example, when you need to convert \(-216.35^{\circ}\) into radians, simply plug the degree measure into the equation:
\(-216.35 \times \frac{\pi}{180}\).
This step-by-step conversion process not only enhances understanding but also ensures accuracy, especially when the exercise requires the result to be rounded to three decimal places.
Other exercises in this chapter
Problem 67
You are standing 45 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86 th floor (the observatory)
View solution Problem 67
Verify that \(\cos 2 t \neq 2 \cos t\) by approximating \(\cos 1.5\) and \(2 \cos 0.75\)
View solution Problem 67
The number of hours \(H\) of daylight in Denver, Colorado on the 15 th of each month are: \(\begin{array}{llll}1(9.67), & 2(10.72), & 3(11.92), & 4(13.25), & 5(
View solution Problem 67
Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \cot (\arctan x) $$
View solution