Math 20-1 Chapter 2 Trigonometry Teacher Notes 2.1B Angles in Standard Position

2.1B Angles in Standard Position Exact Values Math 20-1 Chapter 1 Sequences and Series 2.1B Angles in Standard Position Exact Values 2.1.1

Angles in Standard Position Chapter Identify the angles sketched in standard position. Check answer 2.1.2

Torso Angle - Fast Torso angle is very dependent upon the cyclists choice of performance and comfort. A lower position is more aerodynamic as frontal surface area is reduced. 30° to 40° is a good compromise of performance and comfort but does rely on reasonably good flexibility to lower back and hamstrings. Torso Angle - Touring A more relaxed torso angle will take the pressure off the lower back, hamstrings and the neck and distribute loads from hands to seat. 40° to 50° is a suitable angle for longer distances where comfort is the priority over speed. 2.1.3

Reference Angles Determine the measure of the reference angle. Angle in Standard Position (θ) Quadrant Reference Angle (θR) 165° 320° 250° 60° II 15° IV 40° III 70° I 60° Reference Angle (θR) Quadrant Angle in Standard Position (θ) 85° III 46° I 37° IV 52° II Determine the measure of the angle in standard position. 265° 46° 323° 128° 2.1.4

A ship is sailing in a direction given by the bearing N35°E. Sketch the angle. 35° 55° What is the measure of the angle in standard position? 55° What is the measure of the reference angle of the angle in standard position? 55° 2.1.5

opposite hypotenuse adjacent The Primary Trigonometric Ratios Trigonometry compares the ratios of the sides in a right triangle. The Primary Trigonometric Ratios Opposite the angle. There are three primary trig ratios: Opposite the right-angle sine cosine tangent hypotenuse opposite adjacent Next to the angle 30º 1 2 2.1.6

Trig Equations sin 30º= trig function angle trig ratio Knowing the measure of the reference angle, can you label the triangle? 300 2.1.7

Exact Values for Trig Ratios of Special Angles c2 = a2 + b2 22 = a2 + 12 22 - 12 = a2 √3 = a 300 - 600 - 900 600 300 2 2 2 600 600 600 2 1 450 - 450 -900 c2 = a2 + b2 = 12 + 12 = 2 c = √ 2 450 1 450 1 2.1.8 1

Exact Values of Trig Ratios 2.1.9

What do the angles have in common? Quadrant Sin Cos Tan 30° 150° 210° 330° I II III IV What do the angles have in common? What do notice about the ratios of the lengths of sides? Make a conjecture to determine the sign of the trig ratio for each quadrant. 2.1.10

Use your conjecture to determine the sign of the trig ratio for each quadrant. Angle Quadrant Sin Cos Tan 60° 120° 240° 300° I II III IV Angle Quadrant Sin Cos Tan 45° 135° 225° 315° I II III IV 2.1.11

Calculate the horizontal distance to the midline, labeled a. Allie is learning to play the piano. Her teacher uses a metronome to help her keep time. The pendulum arm of the metronome is 10 cm long. For one particular tempo, the setting results in the arm moving back and forth from a start position of 60° to 120°. What is the exact horizontal distance the tip of the arm moves in one beat? Calculate the horizontal distance to the midline, labeled a. a Which trig ratio would you use to determine the length of side a? The exact horizontal distance is 10 10 cm. 60° a 2.1.12

Two angles meet at a dance. Bad Math Jokes: Two angles meet at a dance. How did Mr. 150° get Miss 30° to say yes to a dance? We share the same sine, we were made for each other. Who should Mr. 300° ask to dance? 2.1.13

State the value of each ratio. Using Exact Values Homework State the value of each ratio. 1. sin 300 = 2. cos 450 = 4. sin 600 = 3. tan 450 = 5. sin 1500 = 6. cos 1200 = RA = 300 RA = 600 7. tan 1350 = 8. tan 1200 = RA = 450 RA = 600 9. sin 1350 = 10. cos 1500 = RA = 450 RA = 300 2.1.14

Assignment Suggested Questions Page 83: 8, 9, 13, 16, 17b, 24a,b 2.1.15