Algebra2/Trig: Trig Unit 2 Packet In this unit, students will be able t: Learn and apply c-functin relatinships between trig functins Learn and apply the sum and difference identities Learn and apply the duble-angle identities Learn and apply the ½-angle identities Name: Teacher: Pd: 1
Table f Cntents Day 1: Cfunctins SWBAT: Knw and apply the c-functin relatinships between trignmetric functins Pgs. 4 7 in Packet HW: Pgs. 8 9 in Packet Day 2: Pythagrean s Identities SWBAT: Use Pythagrean s Identities t simplify trig expressins and find functin values Pgs. 10 14 in Packet HW: Pg. 15 in Packet #3, 5, 6,7,8,10,11, 12, 14, 16, 21, 22, 23, 25, 26, and 28 Day 3: Sum and Difference f Angles Identities SWBAT: Find trignmetric functin values using sum, and difference frmulas Pgs. 16 20 in Packet HW: Pgs. 21 23 in Packet Day 4: Duble Angle Identities SWBAT: Find trignmetric functin values using sum, difference, duble, and half angle frmulas Pgs. 24 27 in Packet HW: Pgs. 28 30 in Packet Day 5: Half Angle Identities SWBAT: Find trignmetric functin values using sum, difference, duble, and half angle frmulas Pgs. 31 34 in Packet HW: Pgs. 35 36 in Packet Day 6: Review SWBAT: Find trignmetric functin values using sum, difference, duble, and half angle frmulas HW: Pgs. 37 41 in Packet Day 7: Test 2
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Warm - Up Day 1 C-functins 4
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Summary/Clsure Exit Ticket 7
Day 1 - Hmewrk 8
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Day 2 - Pythagrean Identities SWBAT: use Pythagrean Identities t (1) simplify trignmetric expressins (2) find functin values Warm - Up Given the unit circle with equatin x 2 + y 2 =1, we knw x = and y =. Therefre, (cs ) 2 + (sin ) 2 = 1 We can write (cs ) 2 as cs 2 and (sin ) 2 as sin 2. We can rewrite the abve equatin as cs 2 + sin 2 = 1. This equatin is called an identity. An identity is an equatin that is true fr all values f the variable fr which the terms f the variable are defined. Specifically, the abve identity is called a Pythagrean Identity since it is based n the Pythagrean Therem. 10
Example: Verify that cs 2 π + 3 sin2 π = 1 3 Nw take the Pythagrean Identity cs 2 + sin 2 = 1 Divide it thrugh by cs 2 Divide it thrugh by sin 2 11
Rules f multiplicatin, divisin, additin and subtractin can be applied: Example 2: Simplify by factring: cs 2 + cs = Example 3: Simplify by factring: 1 sin 2 = Example 4: Simplify 1 cs sin cs 1 Example 5: 12
Example 6: Express sec ct as a single functin. Example 7: Write the expressin 1 + ct 2 in terms f sin, cs, r bth. Example 8: Shw that (1 cs )(1 + cs ) = sin 2. Example 9: a) If cs = 1 and is in the furth quadrant, use an identity t find sin. 3 b) Nw find: 1) tan 2) sec 3) csc 4) ct 13
Example 10: If tan A = 7 3 and sin A < 0, find cs A. Summary/Clsure Exit Ticket 14
Day 2 HW #3, 5, 6,7,8,10,11, 12, 14, 16, 21, 22, 23, 25, 26, and 28 15
Day 3: Sum and Difference Frmulas fr Sine and Csine SWBAT: find trignmetric functin values using sum, and difference frmulas Recall that lgarithms dn t distribute the way peple wuld THINK they wuld: lg AB lg A lg B (d yu remember what it is?) Sines and csines dn t distribute like yu think, either. With yur calculatr, prve that sin(a + B) sin A + sin B when A=30 and B=45. Cncept 1: Sum and Difference f Angles 16
Example 3: Find tan (A + B) if tan A = 3 and tan B = 1 2 Cncept 2: Cndensing a sum r Difference 2) 3) tan 47 tan 17 4) 1 + tan 47 tan 17 17
Cncept 3: Using special Angles t rewrite a given angle Example 2: Find the exact value f cs 105 18
Cncept 4: Finding the Sum r Difference with a given Trig Rati Example 3: Example 4: 19
Summary/Clsure: Exit Ticket 2) 20
Day 3 - Hmewrk 1. Find the exact functin value f cs135 by using cs(90 45 ). 2. Find the exact functin value f cs195 by using cs(135 60 ). 3. If 1 cs( A 30 ), then the measure f A may be 2 (1) (2) (3) 30 (2) 60 90 (4) 120 4. Find tan (A B) if tan A = 4 3 and tan B = -8 5. The value f (cs67 )(cs23 ) (sin67 )(sin 23 ) is (1) 1 (2) 2 2 (3) 2 (4) 0 2 6. Find the exact value fsin 75 by evaluating sin(45 30 ). 12 7. If B is acute and sin B, find the value f sin( 90 B). 13 7 3 8. If sin x = and cs y =, and x and y are psitive acute angles, find tan (x + y). 25 5 21
9. The expressin tan ( 180 y ) is equivalent t (1) tan y (2) tan y (3) 0 (4) -1 4 4 10. If sin x,cs y, and x and y are the measures f angles in the first quadrant, 5 5 find the value f sin( x y). 11. The expressinsin 40 cs15 cs 40 sin15 is equivalent t (1) sin 55 (2) sin 25 (3) cs 55 (4) cs 25 3 5 12. If sin A, A is in Quadrant I, cs B,and B is in Quadrant II, find cs( A B). 5 13 12 4 13. If sin x, x is the measure f an angle in Quadrant III, cs y, and y is the measure f an angle 13 5 in Quadrant II, find cs( x y). 14. Find the exact value f cs105 by using cs(135 30 ). 22
12 15. If sin A, A is in Quadrant III, 13 cs( A B). 4 sin B, and B is in Quadrant II, find 5 16. The expressin cs 30 cs12 sin30 sin12 is equivalent t (1) (2) (3) cs 42 (2) cs 42 sin 42 (4) cs 18 cs 2 42 sin 2 42 17. The expressin sin( x) is equivalent t 6 (1) 1 3 sin x (2) sin x 2 2 3 1 1 3 (3) cs x sin x (4) cs x sin x 2 2 2 2 18. If sin( A 30 ) cs60, the number f degrees in the measure f A is (1) 30 (2) 60 (3) 90 (4) 120 1 4 19. If x and y are the measures f psitive acute angles, sin x, andsin y, then sin( x y) equals 2 5 3 4 (1) 10 3 (2) 3 4 10 3 (3) 12 3 12 (4) 43 25 4 25 20. Find tan (A + B) if angle A is in the secnd quadrant, sin A = 0.6, and tan B = 4. 23
Day 4: Duble Angle Identities SWBAT: find trignmetric functin values using duble angle frmulas Warm - Up What are the sine, csine, and tangent ratis? If cs θ = 8 17 and sin θ > 0, what is the value f tan θ? Name 3 sets f Pythagrean s triples? 1),, 2),, 3),, Lessn: What is a duble-angle functin? Where can yu find the duble-angle Identities? 24
Cncept 1: Sine Duble-Angle Identity Mdel Prblem Step 1: Create a right triangle Student Prblem If θ is an acute angle such that sin θ = 3 5, what is the value f sin 2θ? Step 2: Find the rati fr = Step 3: plug int duble-angle Frmula Mdel Prblem 1 If cs A and A is in Quadrant III, express, in 3 fractinal frm fr sin 2θ? Student Prblem 4 If cs A and A is in Quadrant II, express, in 7 fractinal frm fr sin 2θ? Step 1: Create a right triangle Step 2: Find the rati fr = Step 3: plug int duble-angle Frmula 25
Cncept 2: Csine Duble-Angle Identity Mdel Prblem If θ is an acute angle such that cs θ = 3 4, what is the value f cs 2θ? Step 1: Decide which csine duble angle frmula t use Student Prblem If θ is an acute angle such that sin θ = 2, what is the value f cs 2θ? 3 Step 1: Decide which csine duble angle frmula t use Step 2: plug int crrect duble-angle Frmula Step 2: plug int crrect duble-angle Frmula Cncept 3: Tangent Duble-Angle Identity Mdel Prblem 4 If cs A and A is in Quadrant I, find the psitive 5 value f tan 2A. Step 1: Create a right triangle Student Prblem 12 If sin A, where 270 < A < 360 find the 13 value f tan 2A. Step 1: Create a right triangle Step 2: Find the rati fr = Step 3: plug int duble-angle Frmula Step 2: Find the rati fr = Step 3: plug int duble-angle Frmula 26
SUMMARY Exit Ticket 27
Day 4 - Hmewrk 1.Write the identity fr sin 2x = 2. Write the identity fr the cs 2x in terms f: a. sin x and cs x b. cs x nly c. sin x nly 3. Write the identity fr tan 2x = 24 4. If cs A and A is in Quadrant III, express, in fractinal frm, each value: 25 a. sin A b. cs 2A c. sin 2A d. tan 2A 28
3 5. If sin A and A is in Quadrant III, find: 5 a. sin 2A b. cs 2A c. tan 2A d. The quadrant in which 2Aterminates 1 6. If cs A and A is acute, find 3 a. sin 2A b. cs 2A c. tan 2A 7. If cs sin, then cs 2 is equivalent t (1) 1 (2) 0 (3) 2 cs 2 (4) 2 sin 2 29
2 8. The expressin (sin x cs x) is equivalent t 2 2 (1) 1 (2) sin x cs x (3) 1 cs2x (4) 1 sin 2x 9. If sin is negative and sin 2 is psitive, then cs (1) Must be psitive (3) Must be 0 (2) Must be negative (4) May be psitive r negative 7 10. If tan and is a secnd quadrant angle, find: 3 a. sin 2 b. cs 2 c. tan 2 11. If sec = 13 and is in the furth quadrant, find tan 2. 2 12. If = 225, find tan 2. 30
Day 5: Half Angle Identities SWBAT: find trignmetric functin values using half angle frmulas Warm - Up What are the sine, csine, and tangent ratis? If θ is lcated in Quadrant II, such that sin θ = 24, what is the value f tan θ? 25 If θ is an acute angle, such that sin θ = 5, what is the value f sin 2θ? 13 If cs θ = 3 5, what is the negative value f sin 1 2 θ? Lessn: What is a half-angle functin? Where can yu find the half-angle Identities? 31
Mdel Prblem Student Prblem If cs θ = 1, what is the negative value f 9 sin 1 2 θ? Step 1: plug int half-angle Frmula If cs θ = 4, what is the negative value If cs θ = 5 f tan 1 θ? tan 1 θ? 2 2 5 13, what is the psitive value f 32
Mdel Prblem 24 If tan A and A is in Quadrant III, find the 7 psitive value f sin 1 A. 2 Step 1: Create a right triangle Student Prblem If sin. 6 A and A is in Quadrant I, find the negative value f cs 1 A. 2 Step 1: Cnvert.6 t a fractin. Step 2: Create a right triangle Step 2: Find the rati fr = Step 3: plug int half-angle Frmula Step 3: Find the rati fr = Step 4: plug int half-angle Frmula 33
SUMMARY Exit Ticket 34
Day 5 Hmewrk 35
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Day 6 Review f Trig Cncepts/Identities 37
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7 1 16. If x is the measure f a psitive acute angle and cs x, find the value fsin x. 32 2 17. The expressin 1 cs80 2 is equivalent t (1) 1 sin80 2 (2) sin 40 (3) 1 cs 40 (4) cs 40 2 18. If 180 A 270 and sin A = 5 1, find tan A. 3 2 40
Identities Verify the identities belw. 19. 20. Simplify the expressin belw. 21. 22. 23. 24. 41