3.04 - Trig Points, Quadrant Choice, Coterminal Angles

University:
High School
Course:
High School Physics
Academic year:

2024
Views:

318

Pages:

21
Author:

1 1

Trigonometry 3.04: Trigonometric Points, Quadrant Choice, Coterminal Angles Trigonometric Points O Trig Points = Points on the Unit Circle 𝑃 𝜃 = 𝑃 𝑥, 𝑦 = 𝑃(cos 𝜃 , sin 𝜃) Trigonometric Points vs “Regular” Points O Trig Points are on unit circle (r=1): (x, y) where 𝑥 2 + 𝑦 2 = 1 O Points on other circles have coordinates (x,y) where 𝑥 2 + 𝑦 2 ≠ 1 …but trig still works for circles of all radii… Trigonometry for Points on a non-unit Circle Radius = 𝑟 Center = 0, 0 Example O The point (4,3) is the end of a terminal arm of an angle 𝜃. Find the values of sin 𝜃, cos 𝜃 and tan 𝜃 𝑟= 𝑟=5 5 ϴ 4 42 + 32 3 3 sin 𝜃 = 5 4 cos 𝜃 = 5 tan 𝜃 = 3 4 What is the angle? 𝜃 ≈ 36.86989765° 𝜃 ≈ 0.643501108 rad What about…? O The point is (−2, 3) 13 3 ϴ -2 3 13 sin 𝜃 = 13 2 13 cos 𝜃 = − 13 3 tan 𝜃 = − 2 What is the angle? 𝜃 ≈ 56.3099° 𝜃 ≈ 123.6901° ≈ 0.9828 rad ≈ 2.1588 rad 𝜃 ≈ −56.3099° ≈ −0.9828 rad Choosing the Quadrant O When you calculate an angle, there are always TWO answers. Your calculator will only give you ONE. O CALCULATOR says O Ex: O cos 𝜃 = 1 2 𝜋 𝜃 = 60° 𝑜𝑟 𝑟𝑎𝑑 3 O Unit Circle says 𝜋 𝜃 = 60° 𝑜𝑟 𝑟𝑎𝑑 3 5𝜋 𝜃 = 300° 𝑜𝑟 𝑟𝑎𝑑 3 Equivalent Ratios O SINE O COSINE y-coordinates x-coordinates 𝑃(𝜃1 ) 𝑃(𝜃2 ) 𝑃(𝜃2 ) 𝑃(𝜃1) 𝐶𝐴𝑆𝑇 (𝑤ℎ𝑖𝑐ℎ 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒? ) S (sin) A (all) T (tan) C (cos) The Formulas to Find Other Angles 180 − 𝑛 𝜋−𝜃 𝑛 𝜃 180 + 𝑛 360 − 𝑛 𝜋+𝜃 2𝜋 − 𝜃 Alternate chart… 180 − 𝑛 𝜋−𝜃 360 − 𝑛 2𝜋 − 𝜃 𝑛 ± 180 𝜃±𝜋 540 − 𝑛 3𝜋 − 𝜃 360 − 𝑛 2𝜋 − 𝜃 Back to “What about” O The point is (−2, 3) 3 13 sin 𝜃 = 13 2 13 cos 𝜃 = − 13 3 tan 𝜃= − 2 (−2, 3) 180 − 𝑛 𝜋−𝜃 180 + 𝑛 𝜋+𝜃 𝑛 𝜃 360 − 𝑛 2𝜋 − 𝜃 90 < 𝜃 < 180 180 − 56.3099 ≈ 123.6901° What is the angle? 𝜃 ≈ 56.3099° 𝜃 ≈ 123.6901° ≈ 0.9828 rad ≈ 2.1588 rad 𝜃 ≈ −56.3099° ≈ −0.9828 rad 𝜋 <𝜃<𝜋 2 𝜋 − 0.9828 ≈ 2.1588 rad Try This: What is the angle? O The point is (−2, −3) 3 13 13 2 13 cos 𝜃 = − 13 3 tan 𝜃 = 2 sin 𝜃 = − -2 -3 13 ϴ 𝜃 ≈ −56.3099° ≈ −0.9828 rad 𝜃 ≈ 123.6901° ≈ 2.1588 rad 𝜃 ≈ 56.3099° Want Quad 3: ≈ 0.9828 rad 𝜃 ≈ 56.3099° + 180° ≈ 236.3099° ≈ 4.124 𝑟𝑎𝑑 Coterminal Angles When angles share a terminal arm location Try It: Which Quadrant 𝟏. 𝟓𝟕 − 𝟑. 𝟏𝟒 𝟎 − 𝟏. 𝟓𝟕 𝑃 1 I 𝑃(3) II 𝑃(6) IV 𝑃(−2) III 𝑃(15) not as obvious… 𝟑. 𝟏𝟒 − 𝟒. 𝟕𝟏 𝟒. 𝟕𝟏 𝒕𝒐 𝟔. 𝟐𝟖 Finding coterminal angles (ex: 15 rad) O Since 15 is bigger than 6.28, you know this is more than one rotation 1. Divide by 2𝜋 to find HOW MANY rotations 15 ≈ 2.387324146 2𝜋 2. Eliminate the integer value ≈ 0.387324146 3. Multiply the left overs by 2𝜋 to find the radian value of the final rotation portion 0.387324146 (2𝜋) ≈ 2.4336 rad 15 rad and 2.4336 rad Coterminal Angles O Formula or 𝜃 ± 2𝜋𝑡 𝑛 ± 360𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 𝑖𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑢𝑟𝑛𝑠 𝑡∈ℤ Example 1 O Find the Coterminal angle between 0 and 2𝜋 71 𝑟𝑎𝑑 71 2𝜋 ≈ 11.30000096 turns 0.30000096 2𝜋 ≈ 1.884961621 rad Example 2 O Find the Coterminal angle between 0 and 2𝜋 −492 𝑟𝑎𝑑 −492 2𝜋 ≈ −78.304232 turns − 0.304232 2𝜋 ≈ −1.91154604 rad −1.91154604 + 2𝜋 ≈ 4.371639267 rad HW O Visions 2 O P94: 12, 13, 15