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PREC 12 4.2 The Unit Circle and Reference Angles Date: Complete the following table by converting each to degrees. Note the patterns that exist! Arc Length and Sector Angles One of the advantages to measuring angles in radians is for calculating arc lengths. There is a close relation between the arc length and the size of the angle. The general relation can be summarized with the following formula: a = rθ arc length sector angle = circumference angle of circle arc length sector angle = 2π r 2π arc length = 2π r × sector angle 2π a = arc length r = radius θ = sector angle Note: the sector angle must be measured in radians. If it is provided in degrees, you must convert it to units of radian. Example 1: An arc subtends an angle of 1.8 rad. If the radius of the circle is 6 cm, what is the arc length? Example 2: An arc subtends the angle θ . If the radius is 18 cm and the arc length is 90 cm, what is the measure of angle θ to the nearest tenth of a radian? Example 3: Determine the length of the arc of a circle with radius 12m that subtends 240° at the centre. Express the length to 2 decimal places. Pythagorean Theorem and Special Triangles Ratio of Sides: 3-4-5 Right Angle Triangle Ratio of Sides: 5-12-13 Right Angle Triangle Ratio of Sides: Right Angle Isosceles 45o- 45o - 90o Triangle Ratio of Sides: Half an Equilateral 30o - 60o - 90o Triangle Let’s look closely in the first quadrant when we incorporate the special triangles in a UNIT CIRCLE: 30° or π 6 reference angle in Quad 1 What is the coordinate at 0° or 0π ? 45° or π 4 reference angle in Quad 1 60° or π 3 reference angle in Quad 1 And the coordinate at 90° or π 2 ? Example 4: Determine the point on the terminal arm that intersects the unit circle given P (θ ) : a. P ( 76π ) in standard position b. P ( − π4 ) in standard position Example 5: Determine the reference angle and the angle in standard position, in radians, given the following coordinates: a. (− 2 2 , 2 2 ) b. ( 1 2 ,− 3 2 ) Example 6: Determine the reference angle if the angle in standard position is 5.26 radians.