Notes on Measuring the Size of an Angle Radians and Degrees

Transcription

1 Notes on Measuring the Size of an Angle Radians and Degrees The usual way to measure an angle is by using degrees but there is another way to measure an angle and that is by using what are called a radian measure. P Degrees method of measuring an Angle: To describe how we measure angles in degrees you could imagine a circle starting on the x axis at A, which is 0 0, OA is fixed and will be one arm of our angle. You then rotate the second arm in an anti-clockwise direction round the center to a point P we now have an angle AOP A x Using this system one complete revolution of the circle is 60 0 and all other angles are given in terms of this measure. So for example a right-angle which is a quarter turn is 90 0 while in the diagram above the angle AOP is Radian method of measuring an Angle: To measure an angle in radians we again consider a circle of unit length (Radius ) starting on the x-axis at A ( this will be 0 radians ), OA is fixed and will be one arm of our angle. You then rotate the second arm in an anti-clockwise direction round the center to a point P. We then measure the length of the arc AP and this distance will be the angle AOP in radians. P A x So the further along the circumference of the circle you travel the greater the angle. 0 This means that one complete revolution would be the same as walking round the entire circumference of the circle and for a circle of radius unit this distance would be So one complete revolution would be the same as radians. We can compare the two methods used to measure an angle. Since one complete revolution is 60 0 and is also radians we can convert from one angle measure to the other. The most commonly used angles are given below. Angle Radians 60 0 radians 80 0 radians radians radians radians 6 radians radians.

2 A. How to convert Degrees into Radians To convert degrees into radians we use the formula below. Radians degrees 80 Example : Convert 0 0 into radians Solution: Radians degrees radians Example : Convert 0 0 into radians Solution: Radians degrees radians Example : Convert 50 0 into radians Solution: Radians degrees radians Example : Convert 0 into radians (Rounded of to decimal places) Solution: Radians degrees (0.075) 0.0 radians Note: This means that 0 is the same angle as approximately 0.0 radians so you can use this fact to help you visualize what other angles in radians will look like. Also the following angles are useful ones to memorize is the same as radians is the same as radians 70 0 is the same as radians 60 0 is the same as radians Example 5: Convert 7 0 into radians (Rounded of to decimal places) Solution: Radians degrees 7 7 (0.075).7 radians Example 6: Convert 0 0 into radians (Rounded of to decimal places) Solution: Radians degrees 0 0 (0.075) 5. radians 80 80

3 B. How to convert Radians into Degrees. To convert radians into degrees we use the formula below. Degrees Radians 80 Example : Convert radians into Degrees Solution: Degrees Radians Example : Convert 5 radians into Degrees 6 Solution: Degrees Radians Example : Convert 7 radians into Degrees Solution: Degrees Radians Example : Convert radian into Degrees (Rounded of to decimal places) Solution: Degrees Radians (57.0) Example 5: Convert.5 radian into Degrees (Rounded of to decimal places) Solution: Degrees Radians (57.0) Example 6: Convert.5 radian into Degrees (Rounded of to decimal places) Solution: Degrees Radians (57.0).66 0

4 C. How to find the length of an arc. To find the length of an arc S, for an angle θ in radians we use the formula. s rθ Example : Find the length of the arc when θ and r 5 inches Solution: s rθ 5 5 inches The arc length S 5 inches or (.9 inches) Example : Find the length of the arc when θ 0 0 and r miles Solution: We must first convert 0 0 into radians and then we can use the formula. Radians degrees radians s rθ miles or (87.95 miles) The arc length S miles Example : Find the angle θ when the arc length S ft and r 0 ft Solution: s rθ 0θ θ 0 0. θ The angle θ is 0. radians Example : Find the radius r of the arc when θ Solution: s rθ 7 r r r and S 7 cm. cm r The radius r. cm

5 D. How to find the area of a sector. To find the area of a Sector A, for an angle θ in radians we use the formula. A Example : Find the Area of the sector when θ and r 5 inches Solution: A A 5 5 A inches 8 The area of the sector A 5 8 square inches or 9.8 square inches. Example : Find the Area of the sector when θ radians and r miles Solution: A A A sq miles The area of the sector A square miles. Example : Find the angle θ when Area of the sector A 5 sq cm and r.5 cm Solution: A 5 (.5) θ 5.5θ 5 θ.5.8 θ The angle θ.8 radians Example : Find the radius r, when the angle θ radians and the Area of the sector A sq ft Solution: A r r r 7 r 7 r The radius r 7 ft or r 8.9 ft

6 E. Applications. The length of the minute hand is 6 cm, What distance does the tip of minute hand of a clock turn in 5 minutes, when Find the angle for 5 minutes and convert to radians θ θ 90 You can then use the formula for arc length s rθ 6. cm 80 radians. What is the area of a piece of a cake, with an angle of 5 0 and a radius of 5 inches. Convert 5 0 to radians θ 5 radians 80 You can then use the formula for the area of a sector A A 9.8 sq inches. What is the linear speed of a wheel of radius 5 m if it takes 0 seconds to turn radian. Distance is rθ 5. 5 mm We know the Time 0 seconds Linear Speed Distance Time m/sec. How many revolutions in minute will a wheel with a radius 6 inches make if it is travelling at a speed of 5 mph? Distance travelled by one revolution of the wheel C D (6) 8.67 inches Speed 5 miles per hr 5(580)() inches in hour,7,600 inches in hour 7600 inches in minute inches in minute 60 6,960 inches in minute Number of revolutions Number of revolutions Distance travelled in minute Circumference of wheel 6, (approx)

7 5. A school baseball field is in the shape of a sector of a circle as shown. Given that O is the centre of the circle, calculate: the perimeter of the playing field. Solution: In order to find the area of this sector we need to Convert the angle 80 0 into radians O 80 o R radians Degrees m Arc Length rθ Arc Length The shape of the material used to make a tent is a sector of a circle, as shown. O is the centre of the circle. OA and OB are radii of length metres. Angle AOB is 0 A Calculate the area of this piece of material. Solution: In order to find the area of this sector we need to Convert the angle 0 0 into radians. 0 O R radians Degrees B Area of sector Area of sector () 6 6

8 7. The shape opposite is the sector of a circle, centre P, radius 0ft. The area of the sector is 50 square feet. Find the length of the arc QR. Solution: Since we are told that the area of the sector is 50 sq ft and that the radius of the circle is 0 ft we can find the missing angle at the center θ in radians. P θ 0ft Q Area of sector 50 (0) θ 0ft R 50 00θ θ θ Since we now know θ 5 we can now get the length of the arc QR. Arc Length rθ ft 8. The length of the arc of a circle of radius cm is 8 cm what is the Area of the sector? Solution: Since we are told the length of the arc and the radius of the circle we can use this information to find the missing angle at the center θ in radians. Arc Length rθ 8 θ 8 θ θ Since we now know θ we can now get the area of the sector. Area of sector () sq cm