HW. Pg. 334 #1-9, 11, 12 WS. A/ Angles in Standard Position: Terminology: Initial Arm. Terminal Arm. Co-Terminal Angles. Quadrants

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1 MCR 3UI Introduction to Trig Functions Date: Lesson 6.1 A/ Angles in Standard Position: Terminology: Initial Arm HW. Pg. 334 #1-9, 11, 1 WS Terminal Arm Co-Terminal Angles Quadrants Related Acute Angles eg. Find co-terminal angles to 80º B/ Converting Degrees to Radians and back: eg. a) 45º b) 75º c) d) C/ Finding Arc Length eg. Find the arc length if the radius is cm and the angle is 36º.

2 MCR 3UI More Angles in Standard Position Date: Lesson 6. A/ Finding Trig Values for Obtuse Angles Ratio/Angle 50º 130º 70º 110º 15º 165º sin cos tan HW. Pg. 81 #1-6, 10, 11 Do you notice any patterns? The value of sin is. The value of cos is. The value of tan is. What is the connection between and its partner? What about other Quadrants? sin 50º = cos 50º = tan 50º = sin 130º = cos 130º = tan 130º = sin 30º = cos 30º = tan 30º = sin 310º = cos 310º = tan 310º = How do the angles relate? B/ Special Triangles

3 Ratio/Angle 45º 135º 30º 150º 60º 10º sin cos tan eg.1. If = and <A is obtuse, find the other primary trig ratios.. If the point (, 5) lies on the terminal arm find the trig ratios and the angle. 3. Find the angle: a) = b) = c) = 0.46

4 MCR 3UI Graphing Trig Functions Date: Lesson 6.4 Vertical Transformations: (Stretches and Compressions) These transformations take the form = () and will cause the value of y to either stretch or compress depending on a. i) if a > 1 then there will be a of the values by a factor of. ii) if 0 < a < 1 then there will be a of the values by a factor of. iii) if a < 0 then there will be a in the axis. HW. Pg. 374 #1-1 Example: i) Sketch the graph = for ii) Sketch the graphs =, = 3, = iii) Fill in the chart for ii) and note any invariant points. Function Amplitude Period Max Min Domain Range

5 Horizontal Transformations: These transformations take the form = ($) and will cause the value of x to either stretch or compress depending on a. i) if k > 1 then there will be a of the values by a factor of. ii) if 0 < k < 1 then there will be a of the values by a factor of. iii) if k < 0 then there will be a in the axis. Example: i) Sketch the graph = for ii) Sketch the graphs = cos (), = cos ( ) iii) Fill in the chart for ii) and note any invariant points. Function Amplitude Period Max Min Domain Range

6 More examples: i) Sketch: a) =, b) = sin ( ), c) = 3, d) = sin (), e) = ii) Fill in the chart and note any invariant points. Function Amplitude Period Max Min Domain Range

7 MCR 3UI Graphing Trig Functions Date: Lesson 6.5 Vertical Translations: These translations take the form = () + + (Stretches and Compressions) i) if q > 0 then there will be a. ii) if q < 0 then there will be a. This is called a. Example: i) Sketch the graph = for ii) Sketch the graphs = +, = 1, = + 3 iii) Fill in the chart for ii). HW. Pg. 387 #1-11 Function Amp Period Max Min Vertical Shift V. S. Domain Range

8 Horizontal Translations: These translations take the form = (,) i) if p > 0 then there will be a. ii) if p < 0 then there will be a. This is called a. Example: i) Sketch the graph = for ii) Sketch the graphs = cos ( + 45), = cos ( 60), = cos ( + 30) iii) Fill in the chart for ii). Function Amp Perio d Max Min V. S. Phase Shift P.S. Domain Range

9 More examples: i) Sketch: a) = cos( 90) + 3, b) = sin( + 45) 1, c) = cos( + 30) + 1, ii) Fill in the chart. Function Amp Period Max Min V. S. P.S. Domain Range

10 MCR 3UI Graphing Trig Functions Date: Lesson 6.6 (Stretches and Compressions) HW. Pg. 374 #1-1 sin y = a k p ( θ )) + q cos Function Amp Per Max Min P.S. V.S. D R Examples: 1 1. y = 3 (sin(( θ 30)) + 1. y = 4 (cos( ( θ 45)) y = (sin(3θ + 90) 4. y = (cos( θ + 30) Function Amp Per Max Min P.S. V.S. Domain Range y = 3 (sin(( θ 30)) y = 4 (cos( ( θ 45)) y = (sin(3θ + 90) 1 1 y = (cos( θ + 30) + 1 4

13 MCR 3UI More Angles in Standard Position Date: Lesson 6.7 Recap from before graphing: Terminal arm Initial Arm Angle in Standard Position HW. Pg 348 #1-4, 6,7 Quad I θ Quad II 180 θ Quad III θ Quad IV 360 θ sine cosine tan So all this leads to the CAST rule: Examples: 1. If the point (3, 4) lies on the terminal arm find the trig ratios and the angle.

14 . If the point (-, 6) lies on the terminal arm find the trig ratios and the angle. 3. If the point (-1, -4) lies on the terminal arm find the trig ratios and the angle. 4. If the point (5, -5) lies on the terminal arm find the trig ratios and the angle.

15 MCR 3UI Solving Trig Equations I Date: Lesson 6.8 HW. Pg 408 #1-5 What you need to remember: 3 Primary Trig 3 Secondary Trig Ratios: Ratios: CAST Rule: Special Triangles: 45º, 30º, 60º Special values for sin and cos: 1, -1, 0 1. Solve for 0 θ 360 º a) a) sin θ = b) cos = θ c) cos θ = 1 Steps: 1) ) 3) 4)

16 More examples: Solve for θ, where 0 θ 360. a) sec = 3 θ b) cscθ = 3. Solve for θ, where 0 θ 360 a) sin (1 cosθ) = 0 θ b) (sin θ 1)(sin θ + 3) = 0 θ θ d) cos θ + 3cosθ sinθ = 0 c) (tan 1)(tan 3) = 0

17 MCR 3UI Solving Trig Equations II Date: Lesson 6.9 HW. Pg 408 #3acef Remember: 4 acde 5,6,10,11 3 Primary Trig Ratios, CAST Rule, Special Triangles: 45º, 30º, 60º, Special values for sin and cos: 1, -1, 0 And now a special Trig Identity: Steps: 1) Factor if necessary ) Separate into pieces 3) Draw a picture (CAST or Graph) 4) Find the angles 1. Solve for 0 θ 360 º a) sin sinθ = 0 θ b) cos θ + 3cosθ sinθ = 0 Types of Factoring : c) 3cos + cosθ = 0 θ d) tan θ + tanθ = 0 e) 8sin 10cosθ 11 = 0 θ f) 6cos θ + sinθ = 4

MCR 3UI Proving Trig Identities I Date: Lesson 6.10 HW. Pg 398 # An Identity is a statement that Trig Identity # 1: + =1 Trig Identity # : -. Prove: sin θ + cos θ = 1 sinθ tan θ = cosθ Proofs: 1.

18 MCR 3UI Proving Trig Identities I Date: Lesson 6.10 HW. Pg 398 # An Identity is a statement that Trig Identity # 1: + =1 Trig Identity # : -. Prove: sin θ + cos θ = 1 sinθ tan θ = cosθ Proofs: 1. cos θ tan θ = 1 cos θ. csc θ = 1+ cot θ = 1 cosθ 1+ cosθ sin θ sinθ tanθ cosθ + 1 = 1 1 cosθ 1 sinθ tanθ tanθ + sinθ cosθ + 1 = sinθ tanθ cosθ 1 proof noun \ɑprüf\ : something which shows that something else is true or correct : an act or process of showing that something is true mathematics : a test which shows that a calculation is correct

19 MCR 3UI Proving Trig Identities II Date: Lesson 6.11 HW. Pg 398 #3,4 Trig Identity # 1: + =1 Trig Identity # : -. Proofs: sin θ + cos θ cos θ = 1 Tools:. sin θ 1 1 = tanθ sinθ cosθ tanθ sin 1 cosθ 3. = 1+ cosθ sinθ 4. sec θ (1 + cosθ ) = 1+ secθ 5. 1 cos θ = cos θ tan θ

20 MCR 3UI Review and Practice Questions 1. For the following angles: i) sketch the angle in standard position ii) state a co-terminal angle iii) convert the given angle into the opposite measure (ie. Degrees to Radians) iv) state the exact values of each of the three trig ratios a) 00 0 b) Given the point P(-,5) is on the terminal arm of the angle θ, find the 3 trig ratios in fractional form. Use this information to find the value of θ to the nearest tenth of a degree. 3. Solve for θ, 0 0 < θ < a) cos θ = b) 3tanθ + 5 = 0 c) cos θ + cos θ 1 = 0

21 4. Fill in the chart with the required information. If the item does not apply to a function write N/A in the appropriate space. Sketch the graph. FUNCTIONS AMPLITUDE PHASE SHIFT VERTICAL SHIFT PERIOD MAXIMUM & MINIMUM y = cos 3( θ 45) 1 5. Prove the following trigonometric identities: a) ( 1 + tan θ )(1 cos θ ) = tan θ

Lesson 5.6: Angles in Standard Position

Lesson 5.6: Angles in Standard Position IM3 - Santowski IM3 - Santowski 1 Fast Five Opening Exercises! Use your TI 84 calculator:! Evaluate sin(50 ) " illustrate with a diagram! Evaluate sin(130 ) " Q