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Pre Calculus Notes Chapter 6 Trigonometric Functions Name______________________________________________#________ Created by W.L. Bass, page 1 Extra Notes Created by W.L. Bass, page 2 Ms. Bass Pre-Calculus Trig Chapter 6 Tentative Syllabus Blue Day Blue Date Gold Day Gold Date Tue Jan 3 Wed Jan 4 6.1–6.2 Day 1 Notes (Vocab, Radians, Converting Angles, Special Triangles) Day 1 Notes Pop Quiz p.8 Day 2 Notes… p.9 – 10 Thr Jan 5 Fri Jan 6 6.1–6.2 Day 2: The Unit Circle 6.1–6.2 Practice Day 3 Notes… p.13 – 14 Mon Jan 9 Tue Jan 10 6.1–6.2 Day 3 Practice (Working with the Unit Circle) 6.3 Notes… p.17 – 19 Wed Jan 11 Thr Jan 12 6.3 Practice: Properties of Trigonometric Functions 6.4 Notes… p.22 – 24 Fri Jan 13 Tue Jan 17 6.4 Practice: Graphing Sine & Cosine (without phase shifts) 6.5 Notes… p.26 – 28 In-class Topic Homework (due next class period) NO SCHOOL MONDAY JANUARY 16th Wed Jan 18 Thr Jan 19 6.5 Practice: Graphing Secant, Cosecant, and Tangent 6.6 Notes… p.30 – 32 Fri Jan 20 Mon Jan 23 6.6 Practice: Graphing Cotangent and Sine/Cosine Phase Shifts Make sure all practice is complete Tue Jan 24 Wed Jan 25 Put Together Ch 7 Foldable Chapter 6 Review Day TBD Thr Jan 26 Fri Jan 27 Chapter 6 Test 7.1 – 7.2 Flipped Notes Foldable: Back and Top 4 “windows” Created by W.L. Bass, page 3 Ch 6 Trigonometry 6.1 – 6.2 Day 1 Notes After completing these notes, you should be able to… Understand radians v. degrees Convert between radians and degrees Draw an angle – positive and negative rotations Determine what quadrant an angle lies Determine a reference angle for a given angle Find co-terminal angles Find missing measures in special right triangles Find missing sides in a triangle using the Pythagorean theorem VOCABULARY!! An angle is determined by rotating a ray. The vertex of an angle is the common endpoint. An angle is in standard position when the initial side is the positive x-axis. The initial side of an angle is the side of that doesn’t move. In standard position, it will be the positive x-axis. The terminal side of an angle is the side that moved. The measure of an angle is determined by the amount of rotation A positive angle is generated by a . . . A negative angle is generated by a . . . Co-terminal Angles are angles with the same terminal side. Note: Every angle has an infinite number of co-terminal angles. Created by W.L. Bass, page 4 RADIANS… Just another way to measure angles. ONE RADIAN: The portion of the circle (__________________________) that _______________ the radius. Put it together now… Half a circle = ____________ radians and Half a circle = ____________ degrees Helpful Mental Images – Show the indicated portions. 2 2 4 3 6 Created by W.L. Bass, page 5 Ex 1 Converting between degrees and radians a.) Write 50 in radians. b.) Write 3 rad in degrees 8 Ex 2 Sketch, determine its quadrant, find 2 co-terminal angles (positive & negative), find the reference angle a.) 8 rad 15 Coterm__________ & __________Ref__________ b.) 19 rad 12 Coterm__________ & __________Ref__________ Created by W.L. Bass, page 6 Ex 3 Degrees, Minutes, and Seconds Conversions… a.) 1 ________ 1 revolution ________ Convert 70 8'32" to a decimal in degrees b.) 1' ________ Convert 38.628 to D M 'S" Skills You Will Need Next Class 1. Solve for the missing side in the triangle. (Hint: Use the Pythagorean Theorem.) a.) b.) 9 8 4 2. Special Right Triangle Review…Use your special right triangle patterns to find the missing parts in each triangle. DO NOT USE PYTHAGOREAN THEOREM. 45- 45- 90 a.) 30- 60 - 90 and b.) c.) 9 9 14 d.) e.) f.) 21 60 30 20 10 60 Created by W.L. Bass, page 7 POP QUIZ/EXIT TICKET… Did you REALLY understand? Sketch, determine its quadrant, find… the smallest positive & negative co-terminal angles, the reference angle. 1.) 170 Coterm__________ & __________Ref__________ 3.) 5 rad 4 Coterm__________ & __________Ref__________ 2.) 280 Coterm__________ & __________Ref__________ 4.) 7 rad 3 Coterm__________ & __________Ref__________ Created by W.L. Bass, page 8 Ch 6 Trigonometry 6.1 – 6.2 Day 2 Notes After completing these notes, you should be able to… Understand and apply arc length formula Understand and apply area of a sector formula Understand and apply linear and angular speed Use a calculator to evaluate trigonometric functions Understand how the unit circle is developed Know the points, angles, and signs on the unit circle. RECALL FROM GEOMETRY Proof of Arc Length Formula… Circumference Arc Measure Arc Length 1 360 2 r Arc Measure Arc Length 360 1 r Arc Measure Arc Length 180 1 r Arc Measure Arc Length 1 1 180 Thus… S r Thus… A Proof of Area of a Sector Formula… Area Arc Measure Area Sector 360 1 2 r Arc Measure Area Sector 360 1 2 1 r Arc Measure Area Sector 180 2 1 1 1 2 r 2 Trigonometric Functions and Their Reciprocals… sin csc cos sec tan cot Created by W.L. Bass, page 9 Ex 1 Working with arc length a.) If the radius of a circle is 10 cm and the arc length is 18 cm, find the measure of the central angle in radians. Ex 2 Working with area of a sector a.) Find the area of the sector if the radius of a circle is 6 cm and the central angle is 150. Round to hundredths. Ex 3 Linear speed of an Object Traveling in Circular Motion b.) If the radius of a circle is 12 in and the central angle is 120, find the length of the intercepted arc. b.) If a sector has a central angle of 45 and an area of 6 cm2 , find the radius of the circle. Angular Speed (angle traveled over time) t Allows us to convert ___________ to ___________ Linear Speed dist s r rw time t t Suppose Greg is spinning a rock at the end of a 3 foot rope at a rate of 150 revolutions per minute (rpm). a.) Find the linear speed in feet per minute when the rock is released. Ex 4 Use your calculator to evaluate each trigonometric function. Round to 4 decimal places. a.) cot 56 b.) sec 68 b.) Find the linear speed in miles per hour when the rock is released. c.) csc 29 Created by W.L. Bass, page 10 6.1 – 6.2 Day 2 In-Class Notes Building the Unit Circle What do we already know? 45 30 60 Where are we going? 45 30 60 Created by W.L. Bass, page 11 Put It Together – The WHOLE Unit Circle PRACTICE ROUNDS: Section 6.1… p.359 – 361 □ Drawing Angles: #11 – 21 □ Converting DMS: #23 – 33 □ Converting Radians/Degrees: #35 – 69 □ Arc Length: #71 – 77 □ Area of a Sector: #79 – 85 □ Applications: #87, 91, 93, 95, 97, 101 Section 6.2… p.375 – 377 □ Using Your Calculator: #65 – 75 Created by W.L. Bass, page 12 Ch 6 Trigonometry 6.1 – 6.2 Day 3 Notes After completing these notes, you should be able to… Find trig ratios given one trig ratio Find trig ratios given a point on a terminal side of an angle Evaluate trig functions using the unit circle Evaluate reciprocal trig functions using the unit circle Understand “All Students Take Calculus” and the domain/range of trig functions Ex 1 Find the indicated ratios given that is an acute positive angle… a.) Given: sin 7 11 csc cos tan b.) Given: cot 8 5 sin sec tan Ex 2 Let 2, 3 be a point on the terminal side of . Find the indicated ratios. csc cos cot Created by W.L. Bass, page 13 USING the Unit Circle sin cos tan Pythagorean Theorem… Ex 3 Evaluate each of the following. a.) sin 6 b.) cos 3 c.) 7 tan 4 5 sec 4 g.) cos 2 e.) sin 2 h.) 7 sec 6 i.) f.) csc 2 5 csc 3 d.) Created by W.L. Bass, page 14 6.1 – 6.2 Day 3 In-Class Warm-up Fill in the key values… 1.) 2.) 3.) 2 sin 3 7 tan 6 4.) 4 csc 3 8.) 11 cos 6 5.) 3 cot 4 9.) sin 4 6.) 7 tan 3 10.) sec 2 11.) 13 tan 4 3 cot 2 7.) sec 4 Created by W.L. Bass, page 15 12.) 13.) 14.) 4 cos 3 15.) csc 16.) 5 sec 4 19 csc 6 7 cos 4 17.) 13 cot 6 18.) 7 csc 6 19.) 4 sec 3 20.) 5 sin 3 21.) tan PRACTICE ROUNDS: Section 6.2… p.375 – 377 □ □ □ □ □ Using Points to Evaluate Ratios: #13 – 19 Evaluating Angles: #21 – 45 Finding Ratios (Alternate which ratios you find. Do NOT find all 6 each time.): #47 – 63 Using Points on a Terminal Side: #77 – 83 Extensions: #95 – 111 Created by W.L. Bass, page 16 Ch 6 Trigonometry 6.3 Notes After completing these notes, you should be able to… Find trig ratios given one trig ratio and a quadrant restriction Understand how the Pythagorean Identities are developed Understand what trig functions are odd/even Apply odd/even properties to evaluate trig functions Fill in the key values… Review: Evaluate each expression. 1. sec2 tan 2 3 4 Ex 1 Finding ratios… two different ways. a.) Given: 2. is acute & sin Find tan c). Given: & cos Find cot 12 . 13 2 7 . 7 5 sin tan 6 3 3. 7 sec 6 csc 4 3 5 cot 0 & csc . 3 Find cos b.) Given: d.) Given: sin 0 & tan 1 . Find sec Created by W.L. Bass, page 17 Ex 2 Understanding Odd/Even Properties of Trig Functions a.) Look at the parent graphs below for each trig function and determine its type of symmetry. (Rotational around origin or y-axis) Cosine Sine Secant b.) Tangent Cosecant Cotangent Earlier we learned that… even functions have _____________________________ symmetry and f x ______________ . odd functions have _____________________________ symmetry and f x ______________ . Label the functions above as odd or even. c.) sin x ____________ Thus… cos x ____________ tan x ____________ csc x ____________ sec x ____________ cot x ____________ Apply the even/odd properties you just learned to rewrite the expressions below. sin 2 cos 3 tan 4 Created by W.L. Bass, page 18 Ex 3 Understanding Fundamental Identities a.) Complete each of the Reciprocal Identities sin 1 cos 1 tan 1 tan csc 1 sec 1 cot 1 cot b.) Developing the Pythagorean Identities Recall… sin 2 x cos2 x 1 If we divide the equation by sin 2 x , we get… If we divide the equation by cos2 x , we get… sin 2 x cos2 x 1 sin 2 x cos2 x 1 These three identities are collectively called the PYTHAGOREAN IDENTITIES. c.) Using Fundamental Identities to simplify expressions 1. sin 2 1 2. sec2 tan 2 3. cot 2 csc2 4. tan 5. 1 cos cos 6. cos82 tan 82 sin 82 sec2 tan Created by W.L. Bass, page 19 6.3 In-Class Warm-up 1 – 3… Evaluate each expression. 1. 2sin 4 cos 4 6 5 2. 4 tan 2 3 3. 1 cos 2 cos 2 6 3 4 – 6… Finding each ratio. 12 … find sin 5 4. Given: cos 0 & cot 5. Given: 5 tan 0 & sec … find csc 4 6. Given: sin 5 and is an acute positive angle… Find cos 12 Created by W.L. Bass, page 20 7 – 12… Simplify each expression 7. 1 csc2 10. 1 cos2 11. tan 20 8. sin cos cos sin 9. tan 2 sec2 12 sin 20 cos 20 1 sin 2 12 sec2 12 PRACTICE ROUNDS: Section 6.3… p.375 – 377 □ □ □ □ □ □ Applying Co-terminal Angles to Evaluate Angles: #11 – 25 Using “All Students Take Calculus” to Determine Quadrants: #27 – 33 Finding Ratios Given a Ratio (Alternate which ratios you find. Do NOT find all 6 each time.): #35 – 57 Applying Even/Odd Properties: #59 – 75 Simple Trig Identities: #77 – 87 Extensions: #91, 93 Created by W.L. Bass, page 21 Ch 6 Trigonometry 6.4 Notes After completing these notes, you should be able to… Understand all parts of the parent sine curve – Domain, Range, Amplitude, Period, Shape Understand all parts of the parent cosine curve – Domain, Range, Amplitude, Period, Shape Identify all key parts (Domain, Range, Amplitude, Period) of a sine/cosine graph without a phase shift. Graph one period of sine/cosine without a phase shift. Write the equation for a sine/cosine function without a phase shift given a graph. Write the equation for a sine/cosine function without a phase shift given the characteristics. Complete the table… 0 6 4 3 2 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4 11 6 2 sin What would it look like as a graph? Amplitude:__________ Period:______________ Domain:______________ Range:______________ Complete the table… 0 6 4 3 2 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4 11 6 2 cos What would it look like as a graph? Amplitude:__________ Period:______________ Domain:______________ Range:______________ Created by W.L. Bass, page 22 Ex 1 Graph y sin 2 x Vertical Translation: Amplitude: Period: Range: Distance between each “key value” & calculate “key values”… Ex 2 Graph y 3 4cos x Vertical Translation: Amplitude: Period: Range: Distance between each “key value” & calculate “key values”… Created by W.L. Bass, page 23 y B Acos x y B Asin x Vertical Translation Ex 3 a.) Amplitude Period Phase Shift (Horizontal Translation) Writing Equations of Sine and Cosine Functions without Phase Shifts… Period 2 Write the equation of a cosine function, without a phase shift, amplitude = 2, and period = 4 b.) Write the equation for the graph below. Created by W.L. Bass, page 24 6.4 In-Class Warm-up 1. 1 8 Graph y 5 2sin x Vertical Translation: Amplitude: Period: Range: Distance between each “key value” & calculate “key values”… 2. Write the equation of a sine function… without a phase shift x-axis reflection Amplitude = 4 Period = 1 3. Write the equation for the graph below. PRACTICE ROUNDS: Section 6.4… p.404 – 406 □ □ Finding Period and Amplitude: #11 – 19 Matching Practice: #21 – 30 All □ □ Graphing: #35 – 57 Writing Equations: #59 – 71 Created by W.L. Bass, page 25 6.5 Notes Graphs of Tangent and Reciprocal Functions Ex 1 Graphing Reciprocal Functions Recall… sec _______ 2 3 sec 2 _______ csc 0 _______ csc _______ sec _______ csc _______ Graph f x 5sec 2 x … Think:________________________________________________________ Created by W.L. Bass, page 26 Ex 2 Graphing Tangent Functions. 2 4 Evaluate tan for each of the indicated measures. 3 0 4 4 2 5 4 3 2 NOTE: ALL TANGENT GRAPHS WILL INCLUDE TWO PERIODS (i.e. One full revolution) A. What does a tangent graph look like? TANGENT: y tan x B. REFLECTED TANGENT GRAPH: Graphing Tangent 1 3 Graph… f x 2 tan x 1. y tan x 6 Draw a basic curve… Watch for ______________________ & ______________________ 2. Find the “start” for your graph… Set equal to _________ 3. Find the period… Period = 4. Find the distance between the key values… Divide the period into _______ pieces 5. Calculate the “key values” Created by W.L. Bass, page 27 y B A tan x Vertical Translation Amplitude Period Phase Shift (Horizontal Translation) Ex 3 f x 4 tan 3x Reflection or Vertical Translation? Starting point: Period: Distance b/w key values: Calculate the key values: Domain:__________________________________ Created by W.L. Bass, page 28 6.5 In-Class Warm-up 1. Graph f x 2 3csc x … Think:_____________________________________________________ 2. Graph f x 1 3 tan x 4 Reflection or Vertical Translation? Starting point: Period: Distance b/w key values: Calculate the key values: Domain:__________________________________ PRACTICE ROUNDS: Section 6.5… p.414 □ □ Basic Skills: #7 – 15 □ Extensions: #41, 43, 45, 47, 49 Graphing: #17 – 39 Created by W.L. Bass, page 29 6.6 Notes Graphs of Cotangent and Phase Shifts for Sine and Cosine Ex 1 Graphing Cotangent Functions tan _______ 2 Recall… 3 tan 2 _______ cot 0 _______ cot _______ tan _______ cot _______ Graph f x 2cot x 3 Think: Reflection or Vertical Translation? Starting point: Period: Distance b/w key values: Calculate the key values: Domain:__________________________________ Created by W.L. Bass, page 30 Phase Shift (Horizontal Translation) When a sine or cosine graph has a phase shift (horizontal translation), you must first determine your starting point for the graph! To find the starting point, solve… x 0 Ex 2 Graph: f x sin x 4 Vertical Translation: Amplitude: Starting point… Period: Distance between each “key value” Calculate “key values”… Domain:____________________Range:__________________ Created by W.L. Bass, page 31 Ex 3 Graph: f x cos 4 x 2 Vertical Translation: Amplitude: Reflection? Starting point… Period: Distance between each “key value” Calculate “key values”… Domain:____________________Range:__________________ Ex 4 Graph: f x csc 2 x … Think:___________________________________________________ 2 Vertical Translation: Amplitude: Reflection? Starting point… Period: Distance between each “key value” Calculate “key values”… Domain:____________________Range:__________________ Created by W.L. Bass, page 32 6.6 In-Class Warm-up 1. 1 x … Think:_____________________________________________ 2 Graph… f x 1 2sec Vertical Translation: Amplitude: Reflection? Starting point… Period: Distance between each “key value” Calculate “key values”… Domain:____________________Range:__________________ 2. Graph… f x 3 cot 4 x … Think:_____________________________________________ Reflection or Vertical Translation? Starting point: Period: Distance b/w key values: Calculate the key values: Domain:__________________________________Range:__________________ Created by W.L. Bass, page 33 3. Writing Equations with Phase Shifts Recall… y B A sin x and Period Thus… Phase Shift 2 Why? a.) Sine Amplitude: 3 Period: b.) 2 Phase Shift: 2 Cosine – Reflected across x-axis Amplitude: 5 Period: 3 1 Phase Shift: 3 PRACTICE ROUNDS: Section 6.6… p.424 □ □ Graphing Sine/Cosine Phase Shift: #3 – 13 Writing Equations: #15 – 18 All □ Mixed Graphing Review: #19 – 25 (Tangent, Cotangent, Secant, Cosecant) Created by W.L. Bass, page 34