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Scholarship Algebra II Circular Trig Section 1: Angles and Radians We often name angles with Greek letters – the most common are theta (θ) and alpha (α). You can use other Greek letters if you get bored with these. Circular Trigonometry: We will now talk about angles on a circle instead of angles in triangles. This is called standard position. The initial side is always the positive x-axis. initial side The terminal side is the ray where the angle ends. The sign of the angle will indicate direction. Positive angles go counter-clockwise, and negative angles go clockwise. terminal side The positive x-axis is 0 degrees. Coterminal Angles are angles with the same terminal side. The angle at the left could have several different measures. 160° -200° 520° Radians A radian is another way to measure angles. One radian is one radius unit around the circle. An entire circle contains 2π radians (equivalent to 360°). r 2π π = = d 360 180 π ⋅d 180 180 To convert from radians to degrees: d = ⋅r π To convert from degrees to radians: r = Ex. 1: Convert to radians. a) 60° π πR 60 ⋅ = 180 3 c) -300° π 5π R −300 ⋅ =− 180 3 b) 225° π 5π R 225 ⋅ = 180 4 d) 144° π 4π R 144 ⋅ = 180 5 Ex. 2: Convert to degrees. 5π R a) 3 5π 180 ⋅ = 300° 3 π 3π R b) 2 3π 180 ⋅ = 270° 2 π 7π R c) − 12 − 7π 180 ⋅ = −105° 12 π Arc Length How would we find the arc length of a circle with a radius of 4 ft and a central angle of 75°? 75 5 = 360 24 What’s the length around the whole circle? The circumference! C = 2π r = 8π What fraction of the circle is 75°? So the arc length is: s = 5 5π ( 8π ) = ≈ 5.24 ft 24 3 Arc Length Formulas: Degrees: s = θ ⋅ 2π r 360 θ ⋅ 2π r 2π s = θr Radians: s = 5π R Ex. 4: r = 4 cm, θ = 6 Find arc length. Ex. 3: r = 2.8 cm, central angle = 330° Find arc length. s= ⎛ 5π ⎞ 10π s = rθ = 4 ⎜ ⎟ = ≈ 10.5 cm ⎝ 6 ⎠ 3 330 ⋅ 2π ( 2.8 ) ≈ 16.1 cm 360 = 35π , α = 315° Find r. Ex. 5: mAB s= θ ⋅ 2π r 360 35π = 315 ⋅ 2π r 360 r = 35 ⋅ 360 = 20 630