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3. Right Triangle Trigonometry 3.1 3.2 3.3 3.4 3.5 Reference Angle Radians and Degrees Definition III: Circular Functions Arc Length and Area of a Sector Velocities 1 3.1 Reference Angle • • • • Reference Angle Reference angle theorem Calculate trig function using reference angle Approximation 2 3.1 Reference Angle • Definition. The reference angle for any angle θ in standard position is the positive acute angle between terminal side of θ and the x-axis. We denote the reference angle for θ by θˆ ˆ θ θ =θ θ θˆ θˆ θ θ θˆ 3 3.1 Reference Angle Reference Angle Theorem A trigonometric function of an angle and its reference angle differ at most in sign. You need to remember the signs of trig function on each quadrant (Section 1.3, table 1)! 4 3.1 Reference Angle Problems. (1) Draw the given angle in standard position and name the reference angle. (a) 253.8° [4] Ans. 73.8° (b) –330° [10] Ans. 30° (2) Find the exact value (a) cos135° [14] − 2 2 (b) csc300° − [24] 2 3 5 3.1 Reference Angle Problems. (3) Use calculate to find (a) cos238° [30] Ans. –cos(58°) = –0.5299 (b) csc93.2° [36] Ans. csc(3.2°) = 1.0016 (4) Find θ to the nearest tenth of degree, 0° < θ < 360°, and if [50, 62] (a) sinθ = –0.3090 and θ ∈ QIV; (b) secθ = –3.4159 and θ ∈ QII Ans. 342.0° Ans. 107.0° 6 3.1 Reference Angle Problems. (5) Find θ, 0° < θ < 360°, and if • sinθ = − • secθ = 2 and ∈ QIIV 1 2 and θ ∈ QIII [68] Ans. 225° [78] Ans. 315° 7 3.2 Radians and Degrees • • • Radian measure of an angle Conversion between radians and degrees Using calculator 8 3.2 Radians and Degrees • Radian measure of an angle A O θ s O: center of circle θ : central angle r: radius of circle AB: arc s: is the arc length r B s θ (in radians) = r In particular, if s = r, then we get an central angle with exactly one radian. 9 3.2 Radians and Degrees Radian and degree • ⎛ 180 ⎞ 1 (rad) = ⎜ ⎟ ⎝ π ⎠ • 1 = o π 180 (rad) o where π ≈ 3.1415926 Multiply by Degrees Multiply by π 180 Radians 180 π 10 3.2 Radians and Degrees problems (1) Let θ be a central angle in a circle of radius r with arc length s. Find θ if [2, 6] (a) r = 10 in, s = 5 in Ans. 0.5 radians (b) r = 14 cm, s = 1 8 cm Ans. 0.5 radians (2) For each given angle, [14, 16] • Draw the angle in standard position • Convert the angle to radian measure • Name the reference angle to both degrees and radians (a) –120° (b) 390° Ans. ref = 60°, π/3 (rad) Ans. Ref = 30°, π/6 (rad) 11 3.2 Radians and Degrees problems (3) Give the exact value of each of the following. (a) cot π3 [56] (b) sec 56π 1 3 [60] 2 3 (4) For each given angle, [44, 46] • Draw the angle in standard position • Convert the angle to degree measure • Name the reference angle to both degrees and radians (a) 116π (b) − 53π Ans. ref = 30°, π/6 (rad) Ans. Ref = 60°, π/3 (rad) 12 3.2 Radians and Degrees problems (5) Evaluate each expression using x = π6 . Provide exact answer. [74, 78] (a) sin (x − π3 ) − 12 (b) − 1 + 43 cos(2 x − π2 ) −1+ 3 3 8 13 3.3 Definition III: Circular Functions • • • • 3rd definition of trig functions Values of trig function at special angles Domain and range of trig functions Problems 14 3.3 Definition III: Circular Functions If (x, y) is any point on the unit circle, and t is the distance from (1, 0) along the circumference of the unit circle, then the six Trigonometric Functions are defined as below. Trig Function (circular function) sin t = y cos t = x y tan t = x cot t = xy sec t = 1x csc t = 1y y (x, y) t θ =t (1, 0) x 15 3.3 Definition III: Circular Functions (− (− 3 2 , 1 2 1 2 , (− ) 1 2 , 3 2 2π 3 3 2 1 2 π 2 120° 135° 3π ) ) (, ) ( (0, 1) 90° 60° π 3 π 4 4 1 150° 5π 6 2 1 2 45° π 6 180° (–1,0) π (− (− 3 2 ,− 12 1 2 ,− ) 30° ( ) 3 2 , 12 ) 0° 2π 360° (1,0) 7π 6 210° 1 2 1 2 , ) (− 11π 6 5π 4 4π 3 225° 240° 1 2 ,− 3 2 ) 3π 2 270° (0, –1) 5π 3 7π 4 1 2 3 2 ( ( 3 2 ,− 12 1 2 ) 315° ( ,− ) 300° 330° 1 2 ,− ) 16 3.3 Definition III: Circular Functions Domain and range of trig functions Domain of the circular functions (trig function) sin t, cos t all real numbers tan t, sec t all real numbers except t = π/2 + kπ for any integer k cot t, csc t all real numbers except t = kπ for any integer k Range of the circular functions (trig function) sin t, cos t [–1, 1] tan t, sec t all real numbers, or (–∞, ∞) cot t, csc t (–∞, –1]∪[1, ∞), or all real numbers except (–1, 1) 17 3.3 Definition III: Circular Functions Problems (1) Use the unit circle to evaluate each function. (a) cos(225°) [2] (b) cot ( 43π ) − [12] 1 3 2 2 (2) Using the unit circle, find the six trigonometric function for θ = 53π . [16] sin(θ) = − csc(θ) = − 3 2 2 3 cos(θ) = 1 2 sec(θ) = 2 tan(θ) = − 3 cot(θ) = − 1 3 18 3.3 Definition III: Circular Functions Problems (3) Using the unit circle, find all θ between 0 and 2π and for which sin θ = − 12. [22] 7π 11π and 6 6 (4) If angle θ is in standard position and the terminal side of θ intersects the unit circle at the point. [44] (–1/ 10 ,–3/ 10 ), find sinθ, cosθ, and tanθ. sin(θ ) = − 3 10 , cos(θ ) = − 1 10 and tanθ = 3 19 3.3 Definition III: Circular Functions Problems (5) Identify the argument of each function. [54, 58] (a) sin(2A) (b) cos(2 x − 32π ) ans. 2 x − 32π Ans. 2A (6) Use circular function definition to evaluate, [68, 70] (a) cos(2π + π2 ) (b) cos(2π + π3 ) 20 3.3 Definition III: Circular Functions Problems (related to domain and range) (7) Determine if the statement is possible for some real number z ? [74, 76, 80] (a) csc 0 = z (b) sin 0 = z (c) sin z = 1.2 Ans. No Ans. Yes Ans. No 21 3.4 Applications 1) Formula for Arc Length (in a circle) s = rθ r θ must be in radians s θ 2) Formula for Area of a Sector A = 12 r 2θ Again, θ must be in radians r θ s 22 3.4 Applications (1) Find the arc length s of a circle with central angle θ = 30° and radius r = 4 mm. [8] 2π 3 mm (2) Find the radius r of a circle with central angle θ = and arc length s = π cm. [34] 4 3 3π 4 cm (3) The minute hand of a clock is 1.2 cm long. How far does the tip of the minute hand travel in 40 minutes? [12] 1.6π cm 23 3.4 Applications (4) Find the area of the sector formed by central angle θ = 15° and r = 10 m. [46] 25π 6 m2 (5) Example 6. If a sector formed by a central angle of 15° has an area of π/3 square centimeters, find the radius of the circle. 2 2 cm 24 3.5 Velocities r Uniform Circular Motion s Linear velocity: ν = t Angular velocity: ω = θ θ s θ in radians t Relation between two velocities: ν = rω 25 3.5 Velocities (1) Find the linear velocity ν of a point moving with uniform circular motion if the point covers a distance s = 12 cm in time t = 2 seconds. [4] ν = 6 cm/sec (2) Find the distance s covered by a point moving with linear velocity ν = 10 feet per second for a time = 4 seconds. [8] s = 40 ft 26 3.5 Velocities (3) Find the distance s traveled by a point with angular velocity ω = 2 radians per second on a circle of radius r = 4 inches for time t = 5 seconds. [24] s = 40 in (4) Find the angular velocity ω associated with 20 rpm. [30] ω = 40 rad/min 27