3. Right Triangle Trigonometry

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1 . Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities

2 . Reference Angle Reference Angle Reference angle theorem Calculate trig function using reference angle Approximation

3 . 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 θ ˆ θ = θ θ θˆ θˆ θ θ θˆ

4 . 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., table )! 4

5 . Reference Angle Problems. () Draw the given angle in standard position and name the reference angle. (a) 5.8 [4] (b) 0 [0] Ans. 7.8 Ans. 0 () Find the exact value (a) cos5 [4] (b) csc00 [4] 5

6 . Reference Angle Problems. () Use calculate to find (a) cos8 [0] (b) csc9. [6] Ans. cos(58 ) = Ans. csc(. ) =.006 (4) Find θ to the nearest tenth of degree, 0 < θ < 60, and if [50, 6] (a) sinθ = and θ QIV; (b) secθ =.459 and θ QII Ans. 4.0 Ans

7 Problems.. Reference Angle (5) Find θ, 0 < θ < 60, and if sinθ = and θ QIII [68] secθ = and QIIV [78] Ans. 5 Ans. 5 7

8 . Radians and Degrees Radian measure of an angle Conversion between radians and degrees Using calculator 8

9 . Radians and Degrees Radian measure of an angle A O θ s r B O: center of circle θ : central angle r: radius of circle AB: arc s: is the arc length θ (in radians) = s r In particular, if s = r, then we get an central angle with exactly one radian. 9

10 Radian and degree. Radians and Degrees (rad) π = 80 π o = (rad) where π 80 o.4596 Degrees π Multiply by 80 Multiply by 80 π Radians 0

11 . Radians and Degrees problems () Let θ be a central angle in a circle of radius r with arc length s. Find θ if [, 6] (a) r = 0 in, s = 5 in (b) r = cm, s = cm Ans. 0.5 radians 4 Ans. 0.5 radians 8 () For each given angle, [4, 6] Draw the angle in standard position Convert the angle to radian measure Name the reference angle to both degrees and radians (a) 0 (b) 90 Ans. ref = 60, π/ (rad) Ans. Ref = 0, π/6 (rad)

12 . Radians and Degrees problems () Give the exact value of each of the following. π (a) cot [56] (b) sec [60] (4) For each given angle, [44, 46] 5π 6 Draw the angle in standard position Convert the angle to degree measure Name the reference angle to both degrees and radians (a) π (b) 6 Ans. ref = 0, π/6 (rad) 5π Ans. Ref = 60, π/ (rad)

13 problems. Radians and Degrees (5) Evaluate each expression using x = 6. Provide exact answer. [74, 78] (a) sin π ( x ) (b) + 8 π ( x ) π + 4 cos

14 . Definition III: Circular Functions rd definition of trig functions Values of trig function at special angles Domain and range of trig functions Problems 4

15 . Definition III: Circular Functions If (x, y) is any point on the unit circle, and t is the distance from (, 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 x cot t = y sec t = x csc t = y y θ = t (x, y) t (, 0) x 5

16 . Definition III: Circular Functions ( ) ( ), 0, ( ), π 6 π 4 π (0, ) 90 π ( ), ( ) 60, π π 4 45 π 6 0 ( ), 80 (,0) π π 0 60 (,0) 7π 6 ( ) 0 5π, 4 ( ) 5 4π, (, ) 40 π 70 (0, ) 5π 7π 4 00 π 6 (, ) 5 0 ( ), ( ), 6

17 . 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/ + 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,cost [, ] tan t, sec t all real numbers, or (, ) cot t,csct (, ] [, ), or all real numbers except (, ) 7

18 . Definition III: Circular Functions Problems () Use the unit circle to evaluate each function. 4π cot (a) cos(5 ) [] (b) [] () Using the unit circle, find the six trigonometric 5π function for θ =. [6] ( ) sin(θ) = cos(θ) = tan(θ) = csc(θ) = sec(θ) = cot(θ) = 8

19 . Definition III: Circular Functions Problems () Using the unit circle, find all θ between 0 and π and for which sinθ =. [] 7 π and π 6 6 (4) If angle θ is in standard position and the terminal side of θ intersects the unit circle at the point. [44] ( / 0, / 0), find sinθ, cosθ, and tanθ. sin( θ ) =, cos( θ ) = and tanθ = 0 0 9

20 . Definition III: Circular Functions Problems (5) Identify the argument of each function. [54, 58] (a) sin(a) (b) cos( x ans. x Ans. A π ) π (6) Use circular function definition to evaluate, [68, 70] π (a) cos( π + ) (b) cos( π + π ) 0

21 . 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 =. Ans. No Ans. Yes Ans. No

22 .4 Applications ) Formula for Arc Length (in a circle) s = rθ θ must be in radians θ r s ) Formula for Area of a Sector A = r θ Again, θ must be in radians θ r s

23 .4 Applications () Find the arc length s of a circle with central angle θ = 0 and radius r = 4 mm. [8] π mm () Find the radius r of a circle with central angle θ = and arc length s = π cm. [4] π 4 4 cm () The minute hand of a clock is. cm long. How far does the tip of the minute hand travel in 40 minutes? [].6π cm

24 .4 Applications (4) Find the area of the sector formed by central angle θ = 5 and r = 0 m. [46] 5 π 6 m (5) Example 6. If a sector formed by a central angle of 5 has an area of π/ square centimeters, find the radius of the circle. cm 4

25 .5 Velocities Uniform Circular Motion θ r s Linear velocity: Angular velocity: ν = Relation between two velocities: s t ω = θ t θ in radians ν = rω 5

26 .5 Velocities () Find the linear velocity ν of a point moving with uniform circular motion if the point covers a distance s = cm in time t = seconds. [4] ν = 6 cm/sec () Find the distance s covered by a point moving with linear velocity ν = 0 feet per second for a time = 4 seconds. [8] s = 40 ft 6

27 .5 Velocities () Find the distance s traveled by a point with angular velocity ω = radians per second on a circle of radius r = 4 inches for time t = 5 seconds. [4] s = 40 in (4) Find the angular velocity ω associated with 0 rpm. [0] ω = 40 rad/min 7

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