Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction. Angle Initial Standard Position of an Angle Ray An angle on the coordinate plane is in standard position if the verte is at the origin and one ray is on the positive -ais. III IV Eample 1: Draw a rotation diagram and identify the quadrant the terminal ray falls in. (a θ = 50 (b θ = 10 (c θ = 400 QI Q Q4 Quadrantal Angles Angles in standard position whose terminal side lies on either the -ais or y-ais are called quadrantal angles. Coterminal Angles Two angles in standard position with the same terminal side are called coterminal angles. Eample : Give a positive and negative angle that are coterminal with the following angles. (a α = 90 (b α = 0 (c α = 10 450 and -70 0 and -690 480 and -40
Reference Angles For any angle θ in standard position whose terminal side lies in one of the four quadrants there eists a reference angle θ. ref A reference angle is the positive acute angle formed by the terminal side of θ and the -ais. Eample : For each of the following angles draw a rotation diagram and then state the reference angle. (a β = 160 (b β = 00 (c β = 10 (d β = 78 0 78 60 0 (e β = 110 (f β = 80 (g β = 605 (h β = 410 80 70 65 50 Radian Measure - Angles can also be measured in units that are based on arc length. Definition: One radian is the measure of an angle in standard position with a terminal side that intercepts an arc with the same length as the radius of the circle. π radians = 60 and π radians = 180 CONVERTING BETWEEN DEGREES AND RADIANS (Formulas given on Reference Sheet To convert from degrees to radians multiply the number of degrees by π radians 180 To convert from radians to degrees multiply the number of radians by 180 π radians Eample 4: Convert each of the following common angles in degrees into radians. Epress your answers in terms of pi. (a θ = 90 (b θ = 10 (c θ = 5 π π 5 π 4
Eample 5: Convert each of the following common radian angles into degrees. 5π π (a θ = (b θ = (c θ = 6 150 70 15 Eample 6: Convert each of the following radian angles (which aren t in terms of π into degrees. Round your answers to the nearest degree. (a θ = 5.8 (b θ = 4. (c θ =.5 41 14 π 4 UNIT CIRCLE
Name all angles [ 0 60 ] who have reference angles of the following angles. 0 45 60 7 180-0=150 180+0=10 60-0=0 180-45=15 180+45=5 60-45=15 180-60=10 180+60=40 60-60=00 180-7=108 180+7=5 60-7=88 THE DEFINITION OF THE SIX TRIG FUNCTIONS For an angle in standard position whose terminal ray passes through the point ( y on the unit circle: ( θ = y csc( θ sin the -coordinate cos tan ( θ = the -coordinate sec( θ ( θ = sin( θ cos( θ = opposite = y coordinate adjacent coordinate = 1 1 sin( θ = = hypotenuse y coordinate opposite = 1 1 cos( θ = the -coordinate = hypotentuse adjacent cos( θ cot ( θ = sin( θ = adjacent = coordinate opposite y coordinate ASTC S T A C 1 1
Evaluate each epression using the Unit Circle (give eact values. Then check on your calculator. 1. sin ( 5. sec ( π. csc ( 45 sin( 45 1 ( cos 180 1 = 1 Reciprocal of sin(45-1 4. sin ( π 5. tan ( 60 6. cot ( π 0 undefined 7. cos ( 10 8. cos( 0 11π cos 6 9. 4π sin + cos( 0 + sin( 60 1 1
All of the points on the unit circle must satisfy the equation for the coordinate 1. Every point on the unit circle must satisfy the equation - coordinate given 5 y =. 1 5 + = 1 1 5 + = 1 169 144 = 169 1 =± 1 THE PYTHAGOREAN IDENTITY + y = 1. Verify that this equation is true 1 + 1 + = 1 4 4 + y = 1. Find all possibilities for the For any angle θ ( θ ( θ cos + sin = 1 EXTENSIONS OF THE UNIT CIRCLE Point P is located at the intersection of the unit circle and the terminal side of θ = 115. Find the coordinates of P to the nearest thousandth. ( cos( sin( ( cos( 115sin( 115 (. 4. 906 Let P( -5 be a point on the terminal side of θ in standard position. Find the eact value of sin ( θ. Draw a reference triangle. 5 5 9 = 9 9 Point P is located at the intersection of a circle with a radius of 4 and the terminal side of θ = 45. Find the eact coordinates of P. ( r cos( r sin( ( 4cos( 45 4sin( 45 4 4 ( The terminal side of θ in standard position is in 1 Quadrant III and csc ( θ =. Find the eact 1 value of cos ( θ. Draw a reference triangle. cos( θ 5 = 1
Practice f = 10 sin then f ( 0 =? 1. If ( ( (1 ( ( 5 (4 4 π f = and g = cos then g f =?. If ( ( ( (1 1 ( ( 0 (4 1. Which of the following represents a rational number? π (1 sin 6 ( π sin π ( cos 4 (4 5π cos 4 4. When drawn in standard position an angle β has a terminal ray that lies in the third quadrant. It is 8 known that cos ( β =. Which of the following represents the value of sin ( β? 17 (1 9 ( 8 17 9 ( 15 (4 7 17 9 5. Which of the following is equal to sin ( 00? (1 sin ( 60 ( sin ( 0 ( sin ( 60 (4 sin ( 0 6. For an angle α it is known that its reference angle has a sine value of 4. If the terminal ray of α when 5 drawn in standard position falls in the third quadrant then what is the value of cos( α? (1 ( 5 4 ( 4 (4 5 5
7. The point E( 7 4 lies on the circle whose equation is + y = 65. If an angle is drawn in standard position and its terminal ray passes through E what is the value of the sine of this angle? (1 7 ( 7 ( 4 (4 4 4 5 8. Find the eact value of each of the following. (a π sin (b 4π cos (c 5π tan (d π sec 9. Which of the following could not be the value of sin ( θ? Eplain how you can tell. (1 11 ( 1 ( 5 4 5 (4 1 10. A person on a Ferris wheel sits a distance of 45 feet from the Ferris wheel s center. If they are at an angle of 10 when measured in standard position then how high above the center of the wheel are they to the nearest foot? (1 9 feet ( 1 feet ( feet (4 feet Answers: 1 ( (4 (1 4 ( 5 ( 6 (1 7 (4 8a. Q II b. Q III 1 c. Quadrant IV d. Quadrant III 9 ( 10 (1