Angles Degree and Radian Worksheet Quick Guide

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Angles Degree and Radian Worksheet

When studying geometry and trigonometry, understanding angles is crucial. Angles are used to measure the amount of rotation between two rays that meet at a common endpoint, known as the vertex. There are two common units used to measure angles: degrees and radians. In this worksheet, we will explore the relationship between degrees and radians, and practice converting between the two units.

Degrees:

Degrees are the most commonly used unit to measure angles. A full circle contains 360 degrees, with each degree representing 1/360th of the circle. When measuring angles in degrees, we use the symbol ‘°’. Angles are measured in a counterclockwise direction, starting from the positive x-axis.

Radians:

Radians are another unit used to measure angles, particularly in trigonometry. A full circle contains 2π radians, with each radian representing the angle subtended by an arc whose length is equal to the radius of the circle. When measuring angles in radians, we use the symbol ‘rad’.

Relationship between Degrees and Radians:

To convert between degrees and radians, we can use the following formulas:

1. To convert from degrees to radians:

Radians = Degrees x (π/180)

2. To convert from radians to degrees:

Degrees = Radians x (180/π)

Example 1:

Convert 45 degrees to radians.

Radians = 45 x (π/180)

Radians = π/4

Therefore, 45 degrees is equivalent to π/4 radians.

Example 2:

Convert 3π/2 radians to degrees.

Degrees = (3π/2) x (180/π)

Degrees = 270

Therefore, 3π/2 radians is equivalent to 270 degrees.

Worksheet:

1. Convert the following angles from degrees to radians.

a) 60 degrees

b) 90 degrees

c) 120 degrees

d) 180 degrees

e) 270 degrees

2. Convert the following angles from radians to degrees.

a) π/6 radians

b) π/3 radians

c) 5π/4 radians

d) 2π radians

3. Find the angle in radians that is equivalent to 135 degrees.

4. Find the angle in degrees that is equivalent to 5π/6 radians.

5. The angles in a triangle are given as 30°, 60°, and 90°. Convert these angles to radians and determine the sum of the angles in radians.

6. A circle has a radius of 5 units. Find the length of an arc that subtends an angle of π/3 radians at the center of the circle.

7. The angular speed of a wheel rotating at 100 revolutions per minute is given as 2π radians per second. Calculate the linear speed of a point on the wheel that is located 0.4 meters from the center.

Answers:

1.

a) Radians = 60 x (π/180) = π/3

b) Radians = 90 x (π/180) = π/2

c) Radians = 120 x (π/180) = 2π/3

d) Radians = 180 x (π/180) = π

e) Radians = 270 x (π/180) = 3π/2

2.

a) Degrees = (π/6) x (180/π) = 30

b) Degrees = (π/3) x (180/π) = 60

c) Degrees = (5π/4) x (180/π) = 225

d) Degrees = (2π) x (180/π) = 360

3. Radians = 135 x (π/180) = 3π/4

4. Degrees = (5π/6) x (180/π) = 150

5. Sum of angles in radians = 30 x (π/180) + 60 x (π/180) + 90 x (π/180)

= π/6 + π/3 + π/2 = 2π/3

6. Arc length = radius x angle = 5 x π/3 = 5π/3 units

7. Linear speed = Angular speed x radius = 2π x 0.4 = 0.8π meters per second

Conclusion:

Understanding angles and the units used to measure them is essential in geometry and trigonometry. Converting between degrees and radians allows us to work with angles in different contexts and solve various problems. This worksheet provides practice in converting between degrees and radians and applying these concepts to calculate angles and lengths in different situations. Practice these exercises to strengthen your understanding of angles and their measurements in degrees and radians.

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