Name___________________________
Hour_________
PRE-CALCULUS
Chapter 7: Trigonometric Identities and Equations
_____
Day 1: p. 427 – 428 # 14 – 16, 25 – 37 odd, 45 – 49 odd
_____
Day 2: p. 434 – 435 # 17 – 28 all (skip #22)
*Due in 2 days
_____
Day 3: Workday on p. 434 – 435
_____
Day 4: Take Quiz 7 – 1
_____
Day 5: p. 442 – 443 # 16 – 20, 27, 29
______
Day 6: p. 454 – 455 # 15, 18, 19 – 29 odd, 30
_____
Day 7: WS #1 (*Do #4 and #7 together)
_____
Day 8: Take Quiz 7 – 2
_____
Day 9: WS # 2 #’s 1 – 6
_____
Day 10: Finish WS #2 (#’s 7 – 12) and Begin Test Review
_____
Day 11: Chapter 7 Test – Part 1 (formulas only)
Continue working on Test Review
_____
Day 12: Chapter 7 Test – Part 2
***For test review, also do the following problem on a separate sheet
of paper: Solve cos 2x = –sinx , for all values of x, 0°  x < 360°.
Pre-Calculus
Lesson #1, Ch. 7: Basic Identities
sec A =
1
csc A =
tan A =
cot A =
sin2A + cos2A = ____ Divide 1 by sin2A:
Divide 1 by cos2A:
Solve for sin2A:
sin2A =
; Now factor this:
Solve for cos2A:
cos2A =
; Now factor this:
While we’re at it: Factor sin2x – 2sinx – 3
Simplify: csc  cos  tan 
Multiply (sinx + cosx)2
sin 2 A
1  cos A
Simplify sinxtanx + cosx to get secx
2
3
If cos   ,  in Q4, find csc 
HW: p.427-428 #14-16, 25-37 odd, 45-49 odd
tan 2   sin 2 
tan 2  cos 2 
Pre-Calculus
Lesson #2, Ch. 7: Verifying Trig Identities
Verify each of the following.
tan 2 x  sin 2 x  tan 2 x  sin 2 x
sec x  1 cos x  1

0
sec x  1 cos x  1
HW: p.434-435 #17-28 all (skip 22). (Due in 2 days-workday on this tomorrow)
Pre-Calculus
Lesson #3, Ch. 7: Sum and Difference Formulas
Sum and Difference Formulas
sin (A + B) = sinAcosB + cosAsinB
tan (A + B) =
tan A  tan B
1  tan A tan B
tan (A – B) =
tan A  tan B
1  tan A tan B
sin (A – B) = sinAcosB – cosAsinB
cos (A + B) = cosAcosB – sinAsinB
cos (A – B) = cosAcosB + sinAsinB
Find the exact value of each.
1. sin 75º
We cannot use our calculators to find the exact value, so we are going to break 75º into a
sum of two special angles that we’ve worked with in the past.
So, sin 75º = sin (
2. cos 165°
3. tan 285º
+
)
*recall- sin (x + y) = sinx cosy + cosx siny
4. Find the exact value if 0 < x <

:
2
cos (x – y) if cos x =
*recall, cos (x – y) = cosx cosy + sinx siny
cos x =
3
5
6.
Prove:
so we will need 4 quantities: cos x, cos y
sin x, sin y
tan y =
5. Find the exact value if 0 < x <


:
2
sin   x  = cosx
2

HW: p.442 – 443 #16 – 20, 27, 29
3
5
, and tan y = .
5
12
5
12
sin (x + y) if cos x =
12
8
and sin y =
.
37
17
Pre-Calculus
Lesson #4, Ch. 7: Double and Half-Angle Formulas
cos 2θ =
If we replace cos2θ in the above with 1 – sin2θ we get
If we replace sin2θ with 1 – cos2θ we get
sin 2θ =
tan 2θ =
The above are called the Double Angle Identities.
Half Angle Identities
sin
1
 =
2
cos
1
 =
2
tan
1
 =
2
1. Find cos 15º using a half-angle formula.
3. Prove cot A 
4. If cos x =
2. Find tan 112.5º using a half-angle formula.
sin 2 A
1  cos 2 A
4
and x is in Q2, find sin 2x, cos 2x, and tan 2x.
5
HW #24: p.454 – 455 #15, 18, 19 – 29 odd, 30. The answer to # 18 is 
2 3
.
2
Day 6, Ch. 7: Quiz Review
Worksheet #1, Ch. 7
Name _______________________________
5
24
and tan  =
, find the exact values of the following.
13
7
 and  are Q1 rotations.
Given that cos  =
1.
2.
csc 
cos 
5.
6.
3.
sin 2 
7.
4.
cot 2 
8.
9.
Use a sum or difference formula to find tan 15°.
10.
Use a half-angle formula to find sin 165°.
sin (  +  )
tan (  –  )
1
cos 
2
1
tan 
2
Verify the following.
11.
sin2x =
2 tan x
1  tan 2 x
12.
2 cos 2 x
 cot x  tan x
sin 2 x
13.
2tan  csc 2 - tan2 = 1
14.
1  cos 2
 cot 2 
1  cos 2
15.
 
tan   = csc  – cot 
2
16.
 
   sin 
sin    cos   =
2
2
2
Answers
1.
13
12
6. 
36
323
2.
7
25
3.
120
169
4. 
527
336
7.
3 13
13
8.
3
4
9. 2 –
3
5.
10.
204
325
2 3
2
Pre-Calculus
Lesson #5, Ch. 7: Solving Trigonometric Equations (day 1)
*Recall: The principal values of sin/tan are in Q1 and QIV (-90º to 90º), cos is in Q1 and Q2 (0º-180º).
Steps to follow when solving trig equations:
1. When necessary, get to one trig function.
2. Never divide by trig functions. If it cancels a trig function you loose answers!
3. Look to factor.
4. Check answers if you square both sides.
Solve each of the following for the principal values of x.
1.
2cos2x – 3cosx + 1 = 0
2.
4sin2x + 5 = 8
Solve for all values of x, 0°  x < 360°.
3.
2cos2x – sinx – 1 = 0
HW: WS #2 #’s 1 – 6
4.
3 cscx + 7 = 0
Pre-Calc
Worksheet #2, Ch. 7
Name _______________________________
Hour ________
Solve the following equations for all values of x such that 0°≤ x < 360°.
1.
2csc x – 5 = 0
2.
3.
sin x cot2x – 3sin x = 0
4.
5.
tan 2x = 3tan x
6.
2tan x = sin x
7 sec x – 4 = 0
3cos x – 2sec x – 1 = 0
7.
cos 2x =
1
6
8.
cos2x – sin2x + sin x = 1
9.
3cos 2x + 2sin2x = 2
10.
2tan2x + 9tanx – 5 = 0
11.
3tan2x = 7secx – 5
12.
2cos4x – 3cos2x + 1 = 0
Pre-Calculus
Lesson #6, Ch. 7: Solving Trigonometric Equations (day 2)
Solve for all values of x, 0°  x < 360°.
1.
cos 2x =
3
4
2.
6sinx – 6cscx = 5
HW: WS #2 #’s 7-12 & Begin Test Review
***For test review, also do the following problem on a separate sheet of paper:
Solve cos 2x = –sinx , for all values of x, 0°  x < 360°.
Test Part 1 (formulas only) – Monday
Test Part 2 – Tuesday