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Chapter 8 Unit Circle and Radian Measures Opening problem 1. Consider an equilateral triangle with sides 2 cm long. Altitude [AN] bisects side [BC] and the vertical angle BAC. A Can you see from this figure that sin 30o =1/2? Use your calculator to find the values of sin 30o , sin 150o , sin 390o , sin 1110o and sin(-330o). 2 30o 30o 60o What do you notice? Can you explain why this result occurs even though the angles are not between 0o and 90o ? B 60o N 1 C 2. Degree, minute, second: One degree : 1o = 1/360 of a revolution One minute : 1’ = 1/60 of a degree One second: 1’’ = 1/60 of a minute 3. 1 radian (1c) ≈ 57.3o Radian is an abbreviation for “radial angle” Degree-radian conversion x 180 Degree Radian x 180 5. Convert to radians, in terms of a. 60o b. 80o c. 315o 6. Convert to degrees: a. /5 b. 3 /4 c. 5 /6 Parts of a circle minor arc radius sector center chord segment major arc *minor if it involves less than half the circle *major if it involves more than half the circle. Arc Length X l O q Y r For q in radians, arc length l = qr For q in degrees, arc length l = q/360 x 2r O q r For q in radians, area of a sector A = ½ qr2 For q in degrees, area of a sector A= q/360 x r2 9. Use radians to find the arc length and the area of a sector of a circle of Radius 9 cm and angle 7/4 Arc Length: l = qr Area of a sector: A = ½ qr2 10. A sector has an angle of 1.19 radians and an area of 20.8 cm2. Find its radius and its perimeter. Area of a sector: A = ½ qr2 Perimeter: P = 11. A sector has an angle of 107.9o and an arc length of 5.92 m. Find its: For q in degrees, arc length l = q/360 x 2r A= q/360 x r2 a. radius b area. 12. Find, in radians, the angle of a sector of: Radius 4.3 m and arc length 2.95 m Arc Length: l = qr 13. (number 8 in Exercise 8B) 14. A nautical mile (nmi) is the distance on the Earth’s surface that subtends an angle of 1 minute (or1/60 of a degree) of the Great Circle arc measured from the center of the Earth. A knot is a speed of 1 nautical mile per hour. a. Given that the radius of the Earth is 6370 km, show that 1 nmi is approximately equal to 1.853 km. b. Calculate how long it would take a plane to fly from Perth to Adelaide (a distance of 2130 km) if the plane can fly at 480 knots. 14. A nautical mile (nmi) is the distance on the Earth’s surface that subtends an angle of 1 minute (or1/60 of a degree) of the Great Circle arc measured from the center of the Earth. A knot is a speed of 1 nautical mile per hour. a. Given that the radius of the Earth is 6370 km, show that 1 nmi is approximately equal to 1.853 km. ((1/60)/360) x 2 (6370) = 1.853 km b. Calculate how long it would take a plane to fly from Perth to Adelaide (a distance of 2130 km) if the plane can fly at 480 knots. (2130/1.853) /480 = time 2.4 hours = time The unit circle is the circle with center (0, 0) and radius 1 unit. (0, 1) (-1, 0) (0, 0) (0, -1) (1, 0) x2 + y2 = r2 is the equation of a circle with center (0, 0) and radius r. The equation of the unit circle is x2 + y2 = 1. q is positive for anticlockwise rotations and negative for clockwise rotations. + P - Definition of sine and cosine SohCahToa P(cos q, sin q) q cos q is the x-coordinate of P sin q is the y-coordinate of P Pythagorean identity of sine and cosine cos2 q + sin2 q = 1. Domain and range of sine and cosine of a unit circle. For all points on the unit circle, -1 < x < 1 and -1 < y < 1. So, -1 < cos q < 1 and -1 < sin q < 1 for all q. (0, 1) (-1, 0) (0, 0) (0, -1) (1, 0) Definition of tangent tan q sin q cos q 22. PERIODICITY OF TRIGONOMETRIC RATIOS For q in radians and k є Z , cos (q + 2k) = cos q and sin (q + 2k) = sin q. For q in radians and k є Z , tan(q + k) = tan q. 24. sin (180 – q) = sin q and cos (180 – q) = -cos q 25. Find all possible values of cos q for sin q = 2/3. Illustrate your answer. 25. Find all possible values of cos q for sin q = 2/3. Illustrate your answer. sin q = opp/hyp 3 2 3 q q -√5 2 0 √5 Use Pythagorean Theorem to solve for the 3rd side. cos q = √5/3 or -√5/3 26. If sin q = -3/4 and < q < 3/2, find cos q and tan q without using a calculator. 26. If sin q = -3/4 and < q < 3/2, find cos q and tan q without using a calculator. the domain for q is in the between and 3/2 puts us in the third quadrant. cos q = -√7/4 -√7 -3 4 tan q = -3/-√7 = (3√7) / 7 27. If tan q = -2 and 3/2 < q < 2, find sin q and cos q. 27. If tan q = -2 and 3/2 < q < 2, find sin q and cos q. cos q = √5/5 sin q = -2√5/5 1 √5 -2 28. If q is a multiple of /2, the coordinates of the points on the unit circle involve 0 and +1. 29. If q is a multiple of /4, but not a multiple of /2, the coordinates involve +√2 / 2. 30. If q is a multiple of /6, but not a multiple of /2, the coordinates involve + ½ and + √3 / 2. 31. Use a unit circle diagram to find sin q, cos q and tan q for q equal to: a. /4 b. 5/4 c. 7/4 d. e. -3/4 31. Use a unit circle diagram to find sin q, cos q and tan q for q equal to: a. a. /4 b. 5/4 c. 7/4 d. e. -3/4 2 2 , 2 b. , 1 2 2 2 , 2 c. 2 , 1 2 2 , 2 , 1 2 d . 0 , 1, 0 e. 2 2 , 2 2 , 1 32. Without using a calculator, evaluate: a. sin2 60o b. sin 30o cos 60o c. 4sin 60o cos 30o 32. Without using a calculator, evaluate: a. sin2 60o a . sin 60 (sin 2 o o 60 )(sin 3 60 ) o 2 b. sin 30o cos 60o b . sin 30 cos 60 o o 3 3 3 c . 4 sin 60 cos 30 4 2 2 o c. 4sin 60o cos 30o 1 1 1 2 2 4 Complete the rest and report out. o 3 2 3 4 33. Equation of a straight line. If a straight line makes an angle of q with the positive x-axis then its gradient is m = tan q