MTH113 Lecture Ch 5.1 Degree vs Radian - Arc Length PDF

Chapter 5.1 Angle Measurement Degrees vs Radians An angle of 360 degrees is one complete revolution. Degree measurement is fairly common knowledge. However, the other form of angle measurement is not common knowledge. In this class, we will use both units of measure. We will use mostly radian measur...

Chapter 5.1 Angle Measurement Degrees vs Radians An angle of 360 degrees is one complete revolution. Degree measurement is fairly common knowledge. However, the other form of angle measurement is not common knowledge. In this class, we will use both units of measure. We will use mostly radian measure in Calculus, so it is important to learn how to "think in radians". In this lesson, we will: > Learn about the Unit Circle > Learn the definition of the central angle > Learn how to convert from degrees to radians > Learn how to convert from radians to degrees > Locate 1 degree and 1 radian on the Unit Circle > Learn the formula for the length of the subtended arc > Solve arc length, radius, and central angle problems  Unit Circle The Unit Circle is fundamental to the study of Trigonometry. It has these characteristics: Its Radius = 1 unit and its Center = origin The vertex of a central angle is at the origin. Each central angle can be measured in either degrees or radians. Note: Usually, only Standard Angles are shown on a Unit Circle.  How to Convert from Degrees to Radians A circle is subdivided into degrees or radians. Standard angles are shown on the Unit Circle with both the degree and its equivalent radian measure. You will need to use a conversion formula to convert a non-standard angle from degrees to radians. Here is the formula: * Multiply by pi/180 * Reduce fraction to lowest terms * DO NOT convert to a decimal unless told to Example 1: Convert 56 degrees to radians.  How to Convert from Radians to Degrees A circle is subdivided into degrees or radians. Standard angles are shown on the Unit Circle with both the degree and its equivalent radian measure. You will need to use a conversion formula to convert a non-standard angle from radians to degrees. Here is the formula: * Multiply by 180/pi * Reduce fraction to lowest terms * DO NOT convert to a decimal unless told to Example 2: Convert 3pi/20 radians.  Degree-to-Radian and Radian-to-Degree Formulas Combined Here are several conversion charts you can use for your conversion problems. There are other charts on the web. Choose the one your prefer. Example Problems: Use the appropriate formula to convert from degrees to radians. Remember to reduce the fractions to lowest terms, but do not convert to a decimal. Leave PI in your answer. Example Problems: Use the appropriate formula to convert from radians to degrees. Usually a fractional degree is converted to a decimal.  How does 1 degree compare to 1 radian? Are they the same? NO Can you locate 1 degree and 1 radian on the Unit Circle? Examine the comparison scale. > Notice 1 degree is just more than 0 degrees. > Notice 1 radian is less than 60 degrees. Using the radian-to-degree conversion formula, 1 radian times 180/pi = 180/pi degrees about equals 57 degrees  Locate 1 degree and 1 radian on the Unit Circle 1 degree 1 radian  The formula for the length of the subtended arc We will use the appropriate formula based on whether the angle is given in degrees or in radians. We will solve arc length, radius, and central angle problems in a circle. Examples involving the subtended arc of a circle: 1 - Find the length of the arc subtended by an angle of radians in a circle of radius 10 inches. Round answer to 3 decimal places. Solution: Find s, given central angle radians and radius r = 10 inches. 2 - Find the length of the arc subtended by an angle of 80 degrees in a circle of radius 15 miles. Round answer to 3 decimal places. Solution: Find s, given central angle degrees and radius r = 15 miles. Examples involving the subtended arc of a circle: 3 - Find the length of the radius of the circle with an arc length of 4 feet subtended by an angle of radians. Give exact answer. Solution: Find r, given central angle radians and arc length s = 4 feet Solve for r: Multiply both sides by 4: Divide both sides by 5pi: Answer radius r = Examples involving the subtended arc of a circle: 4 - Find the measure of the subtended angle in a circle with a radius of 6 feet and an arc length of 12 feet. Specify the answer in radian measure. Solution: Find the central angle, given r = 6 feet and s = 12 feet. Note: the radius and arc lenght must be in the same unit of measure. Solve for theta: Divide both sides by 6: Using this formula, the angle answer will be in radians. Check your understanding..... 1 - Convert 62 degrees to radian measure. 2 - Convert to degree measure. 3 - Find the length of the subtended arc in a circle of radius 12 and a central angle of 9 degrees. 4 - Find the radius of a circle whose central angle of radians has a subtended arc length of 32. solutions are on the next page..... Check your understanding..... SOLUTIONS 1 - Convert 62 degrees to radian measure. 2 - Convert to degree measure. 3 - Find the length of the subtended arc in a circle of radius 12 and a central angle of 9 degrees. 4 - Find the radius of a circle whose central angle of radians has a subtended arc length of 32.