Algebra2/Trig Chapter 13 Packet In this unit, students will be able to: Use the reciprocal trig identities to express any trig function in terms of sine, cosine, or both. Prove trigonometric identities algebraically using a variety of techniques Learn and apply the cofunction property Solve a linear trigonometric function using arcfunctions Solve a quadratic trigonometric function by factoring Solve a quadratic trigonometric function by using the quadratic formula Solve a quadratic trigonometric function containing two functions by using identities to replace one of the functions. Name: Teacher: Pd: 1
Table of Contents Day 1: Solving First Degree Trig Equations SWBAT: Solve First Degree Trig Equations Pgs. 3 7 in Packet HW: Pgs. 8 10 in Packet Day 2: Trig Equations by Factoring SWBAT: Solve Trig Equations by Factoring Pgs. 11 14 in Packet HW: Pgs. 15 17 in Packet Day 3: Unfactorable Trig Equations SWBAT: Solve Second Degree Trig Equations using the Quadratic Formula Pgs. 18 21 in Packet HW: Pgs. 22 23 in Packet ***Quiz after Day 3*** Day 4: Solving Trig Equations With More Than One Function SWBAT: Solve trigonometric equations using reciprocal identities Pgs. 24 28 in Packet HW: Pgs. 29 31 in Packet Day 5: Solving Trig Equations With More Than One Function SWBAT: Solve trigonometric equations using Pythagorean s identities Pgs. 32 35 in Packet HW: Pgs. 36 38 in Packet Day 6: Solving Trig Equations With More Than One Function SWBAT: Solve trigonometric equations using Double-Angle identities Pgs. 39 46 in Packet HW: Pgs. 47 51 in Packet 2
Chapter 13: Sections 1 - Solving First Degree Trigonometric Equations SWBAT: Solve first degree trig equations Warm - Up: Identify trig values of quadrantal angles Sine Cosine Tangent Sin 0/360 = Sin 90 = Sin 180 = Sin 270 = cos 0/360 = cos 90 = cos 180 = cos 270 = tan 0 = tan 90 = tan 180 = tan 270 = In the trig function, what does the symbol represent? How do you solve for if given the equation? Draw and label ASTC. What is the purpose of ASTC? What is a reference angle? What is the purpose of a reference angle? 3
**** Advice for Solving Trigonometry Equations**** 1) See if the trig functions match 2tanA + = tan A versus 2tanA + = cot A 2) Substitute a variable in for the matching trig functions (optional) 2tanA + = tan A 3) Determine if the trig function is positive or negative to see which quadrants you are in (ASTC) tana = versus tana = - 4) In order to find a reference angle, you must perform the inverse of Positive values only! tana = - 5) Use the reference angle to find your answers in the correct quadrants (ASTC) 6) Be aware of the given interval! (restrictions, degrees or radian measure) 0 versus 180 versus 0 4
And, just like the other problems, if the trig function is NOT isolated, isolate it first before you solve for the missing angle. If the problem is given with a domain in terms of, then your answers should be in radians. I suggest doing the problem in degrees first, and then convert to radians. Model Problem 1. Find in the interval that satisfies the equation. Student Problem 2. Find the value of x in the domain that satisfies the equation. 3. Find in the interval that satisfies the equation below: 4. Find in the interval that satisfies the equation below: 5
Reciprocal Trig Equations Model Problem 5. Find in the interval that satisfies the equation below: Student Problem 6. Find the value of x in the domain that satisfies the equation below: Practice: 7. Find, to the nearest tenth of a degree, in the interval that satisfies the equation below: 8. Find in the interval that satisfies the equation below: 6
Summary/Closure Exit Ticket 7
Day 1 HW 8
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Day 2: Using Factoring to Solve Trigonometric Equations SWBAT: solve trigonometric equations by factoring Warm - Up: Concept 1: Factorable 2 nd degree Trig Equations Each of the following are considered quadratic (2 nd degree) trigonometric equations. It should be pretty easy to see why. Algebraic 2 nd Degree Equation Solve for x: Trigonometric 2 nd Degree Equation Solve for to the nearest degree in the interval 0 o 360 o : 11
To solve a quadratic trig equation: Set the quadratic = 0, just like you would any quadratic! Factor the quadratic, but instead of using x s, use sin x or whatever function you re given. Now you have two linear equations. Solve each of them. You will have anywhere up to 5 solutions!! Recall that sine x and cosine x can never have a value >1 or <-1. These values will get rejected as solutions. Example 1: Solve interval in the Example 2: Find all values of x in the interval which satisfies the equation. Factoring Technique: Factoring Technique: Factoring Technique: 12
4. Factoring Technique: Practice 13
Summary/Closure: Exit Ticket: 14
Day 2 - HW 15
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Day 3 - Solving UnFactorable Trig Equations SWBAT: Solve trigonometric equations using the quadratic formula Warm - Up: 1) 2) 18
Quadratics that require the Quadratic Formula Algebraic Equation Example: ( ) ( ) ( )( ) ( ) If asked to the nearest ten-thousandth, use your calculator to evaluate: Trigonometric Equation Example: Find x to the nearest degree in the interval 0 o 360 o : ( ) ( ) ( )( ) ( ) OR REJECT OR Examples: 1. Find to the nearest degree all values of in the interval 0 o 360 o that satisfies: 4 sin 2 2 sin 3 = 0 2. Find to the nearest degree all values of in the interval 0 o 360 o that satisfies: 9 cos 2 6 cos = 3 19
3. 4. Find to the nearest minute all values of in the interval 0 o 360 o that satisfies: 4(1 - ) + 5 + 1 = 0. 20
Summary/Closure: To solve a trigonometric equation that is not factorable: Exit Ticket: 21
Day 3 - HW 22
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Day 4: Trig Equations containing more than one function Using Reciprocal Identities Warm Up: Mini Lesson: 24
Let s Review Reciprocal Identities: What do you notice about the trig functions below ( matching, how to solve, factor, identities etc.)? Case 1: Case 2: Case 3: Technique: Technique: Technique: 25
Model Problem Example 1: Find all values of A in the interval 0 o 360 o such that Student Try It Example 2: Find all values of A in the interval 0 o 360 o such that 2 sin A - 1 = csc A 26
Practice: Find all values of x in the interval 0 o 360 o such that: 1) 2) 27
SUMMARY: Exit Ticket 28
Day 4 Homework 29
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Day 5 - Trig Equations containing more than one function USING PYTHAGOREAN IDENTITIES Warm Up: Match each, but do not solve! (Meaning set up an equation with matching trig functions but do not solve!) 1) 2) Trigonometry Equations: If a trig equation contains more than one function, and the functions cannot be separated out and factored, then you have to convert everything to one equation. One way that this can happen is by using one of the Pythagorean identities. Recall the three Pythagorean Identities: OR OR We will primarily use only the top two rows. 32
Example 1: Find, to the nearest tenth of a degree, all values of in the interval that satisfy the equation. Example 2: Find, to the nearest tenth of a degree, all values of in the interval that satisfy the equation. 33
Example 3: Solve for in the interval 0 o 360 o for cos 2 + sin = 1. Example 4: 34
Summary/Closure Exit Ticket 35
Day 5 Homework 36
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Day 6 - Solving Trig Equations with Double Angle Identities Warm-Up: Examine the following questions below. Write down any observations that you make about the questions or the trig equations (similarities, differences, how to solve etc.). DO NOT SOLVE!!!!!! ) technique: b) 2-1 = technique: c) technique: d) technique: e) technique: 39
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+ = 0 44
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SUMMARY Exit Ticket 46
Day 6 Homework 47
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