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UNIT 7
ANALYTIC TRIGONOMETRY
LESSON 7.1
VERIFYING IDENTITIES
PROOFS!
• I know, we have already done proofs. But…
• Now we are able to transform the left, right, or both
sides of the equation to verify the identity.
• Watch out for conjugates, factoring, Pythagorean
Identities, etc…when working on these proofs.
EXAMPLE:
• Verify the identity below:
• tan 𝑥 − sec 𝑥
2
=
1−sin 𝑥
1+sin 𝑥
HOMEWORK:
• Pages 498 – 499 #’s 1 – 47 odds
BELL WORK:
• Verify the following identities:
• 1) ln sec 𝑥 = − ln cos 𝑥
• 2) log csc 𝑥 + log tan 𝑥 = − log cos 𝑥
• 3) 1 − 𝑐𝑜𝑠 4 𝑥 = 2𝑠𝑖𝑛2 𝑥 − 𝑠𝑖𝑛4 𝑥
LESSON 7.2
TRIGONOMETRIC EQUATIONS
EXAMPLE:
• Find the solutions for the equation sin θ = ½.
• How many solutions are there?
• How do we represent them?
EXAMPLE:
• Find the solutions for the equation tan 𝑚 = −1 given
the following restrictions on m:
• A) m is in the interval from (-π, π)
• B) m is any real number
• C) m < 0
EXAMPLE:
• Solve the equation cos 2x = 0 and express the
solutions in both radians and degrees.
EXAMPLE:
• Solve the equation sin θ tan θ = sin θ.
HOMEWORK:
• Page 511 #’s 1 – 13 odds
BELL WORK:
• Solve the equation 4sin² x tan x – tan x = 0 given the
following restrictions:
• A) x is in the interval from [0,2π]
• B) x is any real number
• C) x < 0
EXAMPLE:
• Solve the equation 2sin²x – cos x = 1.
EXAMPLE:
• Solve the equation 𝑐𝑠𝑐 4 2𝑢 = 4.
• *Warning* This problem is so awesome, it may
cause you to become even more awesomer than
you already are, which may cause an overload of
awesome, which has caused fatalities in the past.
Proceed with caution.
EXAMPLE:
• On the interval from [0°,720°], approximate all the
values of x that would satisfy the following equation.
(Round to the nearest degree.)
• 5 sin 𝑥 tan 𝑥 − 10 tan 𝑥 + 3 sin 𝑥 − 6 = 0
HOMEWORK:
• Page 511 #’s 19 – 35 odds
BELL WORK:
• Using the following restrictions on x, solve:
• sec 𝑥 ∙ csc 𝑥 = 2 csc 𝑥
• A) When x is in the interval from [0,4π]
• B) When x is any real number
EXAMPLE:
• Solve the equation 2𝑐𝑜𝑠 2 𝑥 = 5 cos 𝑥 + 3.
HOMEWORK:
• Pages 511 – 512 #’s 37 – 59 odds
• This assignment will be collected!!!
WORD PROBLEM:
• The number of hours of daylight D(t) at a particular
time of the year is approximated by:
• 𝐷 𝑡 = 2 sin
2𝜋
365
𝑡 − 79 + 12
• with t in days and t = 1 corresponding to January 1st.
On approximately what days of the year is there
exactly 13 hours of daylight?
How many days of the year have 11 or more hours of
daylight?
HOMEWORK:
• Pages 512 – 513 #’s 68, 71, 73, 75b
BELL WORK:
• Verify each identity:
• 1)
sin 𝑥
1−cos 𝑥
= csc 𝑥 + cot 𝑥
• 2) 𝑠𝑖𝑛3 𝑥 + 𝑐𝑜𝑠 3 𝑥 = (1 − sin 𝑥 cos 𝑥)(sin 𝑥 + cos 𝑥)
BELL WORK CONTINUED:
• Solve each equation:
• 1) 2𝑐𝑜𝑠 2 𝑥 + 3 cos 𝑥 = −1
• 2) 𝑠𝑒𝑐 5 𝑥 = 4sec 𝑥
• 3) sin 𝑥 + cos 𝑥 cot 𝑥 = csc 𝑥
BELL WORK:
• Simplify the expressions below (find the exact value)
• 1) cos 45° + cos 30°
• 2) cos(75°)
LESSON 7.3
ADDITION AND SUBTRACTION FORMULAS
ADDITION/SUBTRACTION FOR COSINE
• cos 𝑢 − 𝑣 = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣
• cos 𝑢 + 𝑣 = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣
• How could we use these to find the cos 75°?
ADDITION/SUBTRACTION FOR SINE
• sin 𝑢 + 𝑣 = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣
• sin 𝑢 − 𝑣 = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣
• How could we use this to find the sin
13𝜋
?
12
ADDITION/SUBTRACTION TANGENT
• tan 𝑢 + 𝑣 =
tan 𝑢+tan 𝑣
1−tan 𝑢 tan 𝑣
• tan 𝑢 − 𝑣 =
tan 𝑢−tan 𝑣
1+tan 𝑢 tan 𝑣
• How could we use this to find the tan 345°?
COFUNCTION FORMULAS
• Cosine/Sine:
• cos 𝑢 =
𝜋
sin(
2
− 𝑢)
sin 𝑢 =
𝜋
cos(
2
− 𝑢)
• Tangent/Cotangent
𝜋
2
• tan 𝑢 = cot( − 𝑢)
𝜋
2
cot 𝑢 = tan( − 𝑢)
• Secant/Cosecant
𝜋
2
• sec 𝑢 = csc( − 𝑢)
𝜋
2
csc 𝑢 = sec( − 𝑢)
HOMEWORK:
• Pages 522 – 523 #’s 5 – 9 odds, 17 – 21 odds, 35 - 41
BELL WORK:
• If a and b are acute angles such that the csc a =
13/12 and cot b = 4/3, find:
• 1) sin (a + b)
• 2) tan (a + b)
• 3) the quadrant containing a + b
EXAMPLES:
• Verify:
• 1) sin 𝜃 −
3𝜋
2
= cos 𝜃
• 2) tan 𝜋 − 𝜃 = −𝑡𝑎𝑛𝜃
• 3) cos(𝜃 −
5𝜋
)
2
= 𝑠𝑖𝑛𝜃
EXAMPLE:
• Verify:
• 4) cos(𝑢 + 𝑣) ∙ cos 𝑢 − 𝑣 = 𝑐𝑜𝑠 2 𝑢 − 𝑠𝑖𝑛2 𝑣
EXAMPLE:
• Use the addition and/or subtraction formulas to find
the solutions for the equation in the interval from
[0,π].
• sin4x· cosx = sinx· cos4x
HOMEWORK:
• Pages 523 – 524 #’s 10, 22, 26, 36, 38, 40
QUIZ FRIDAY
• Lessons 7.1 – 7.3
• 7.1 Proofs
• 7.2 Solving Trigonometric Equations (either on a
given interval or for all real numbers)
• 7.3 Addition/Subtraction Formulas and Proofs
PRACTICE PROBLEMS:
• Page 524 #’s 54, 55, 58
BELL WORK:
• Verify the following identities:
• 1)
𝑐𝑜𝑡𝑥−𝑡𝑎𝑛𝑥
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥
= 𝑐𝑠𝑐²𝑥 − 𝑠𝑒𝑐 2 𝑥
• 2) ln 𝑐𝑠𝑐²θ = −2ln 𝑠𝑖𝑛θ
CLASS WORK:
• Pages 511 – 512 #’s 31, 34, 38, 42
• Pages 523 – 524 #’s 20, 33, 38, 56
• Also review the proofs from lesson 7.1!
BELL WORK:
• Use the addition and/or subtraction formulas to find
the solutions for the equations in the interval from
[0,2π].
• 1) sin4x· cosx = sinx· cos4x
• 2) tan2x + tanx = 1 – tan2x· tanx
LESSON 7.4
MULTIPLE-ANGLE FORMULAS
DOUBLE ANGLE FORMULAS
• 1) sin 2𝑢 = 2 sin 𝑢 cos 𝑢
• 2) a) cos 2𝑢 = 𝑐𝑜𝑠²𝑢 − 𝑠𝑖𝑛2 𝑢
•
b) cos 2𝑢 = 1 − 2𝑠𝑖𝑛2 𝑢
•
c) cos 2𝑢 = 2𝑐𝑜𝑠²𝑢 − 1
• 3) tan 2𝑢 =
2 tan 𝑢
1−𝑡𝑎𝑛2 𝑢
• Where did these come from???
EXAMPLES:
• Ex1: If the sin x = 4/5 and x is in the first quadrant,
find the exact values of sin2x, cos2x, and tan2x.
• Ex2: Verify the identity, cos 3𝑥 = 4𝑐𝑜𝑠³𝑥 − 3 cos 𝑥.
EXAMPLE:
• Find all solutions:
• 1) sin 2x + sin x = 0
• Verify the identity:
• 2) cos 4𝑥 = 1 − 8𝑠𝑖𝑛2 𝑥 + 8𝑠𝑖𝑛4 𝑥
HALF-ANGLE IDENTITIES
• 1) 𝑠𝑖𝑛²𝑢 =
1−cos 2𝑢
2
• 2) 𝑐𝑜𝑠²𝑢 =
1+cos 2𝑢
2
• 3) 𝑡𝑎𝑛²𝑢 =
1−cos 2𝑢
1+cos 2𝑢
EXAMPLES:
1
8
• Ex3: Verify the identity, 𝑠𝑖𝑛²𝑥 𝑐𝑜𝑠²𝑥 = (1 − cos 4𝑥).
HOMEWORK:
• Page 532 #’s 3, 11, 15, 17, 23
BELL WORK:
• 1) Find the exact value of sin 2x, cos 2x, and tan 2x
if you know that the cscx = -13/5, and x is in the third
quadrant.
• 2) Verify:
𝑠𝑖𝑛2 2𝑥
𝑠𝑖𝑛2 𝑥
= 4 − 4𝑠𝑖𝑛2 𝑥
• 3) Verify: (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥)2 = 𝑠𝑖𝑛2𝑥 + 1
BELL WORK:
• Find all solutions:
• cos 3𝑥 cos 2𝑥 − sin 3𝑥 sin 2𝑥 = −
• Find all solutions between [0,2π]
1
2
• 𝑐𝑜𝑠𝑥 + 3cos( 𝑥) + 2 = 0
1
2
EXAMPLES:
• Verify each identity:
• 𝑐𝑜𝑠2𝑥 =
2−𝑠𝑒𝑐 2 𝑥
𝑠𝑒𝑐 2 𝑥
• 𝑡𝑎𝑛𝑥 + 𝑐𝑜𝑡𝑥 = 2𝑐𝑠𝑐2𝑥
• 𝑡𝑎𝑛𝑥 = 𝑐𝑠𝑐2𝑥 − 𝑐𝑜𝑡2𝑥
CLASS WORK:
• Pages 532 – 533 #’s 2, 4, 18, 20, 25, 35, 37, 40
HALF-ANGLE FORMULAS
• 1)
𝑣
sin
2
• 2)
𝑣
cos
2
• 3)
𝑣
tan
2
=±
1−cos 𝑣
2
=±
1+cos 𝑣
2
=±
1−cos 𝑣
1+cos 𝑣
• Also with tangent:
𝑣
tan
2
=
1−cos 𝑣
sin 𝑣
=
sin 𝑣
1+cos 𝑣
EXAMPLES:
• Find the exact value of the sin 22.5°.
• Find the exact value of the cos 112.5°.
EXAMPLE:
• Find the solution for the equation below that are in
the interval [0,2π].
• 𝑐𝑜𝑠2𝜃 − 𝑡𝑎𝑛𝜃 = 1
• Warning: This one is off the hook…
HOMEWORK:
• Pages 532 – 533 #’s 5, 9, 13, 19, 25, 33, 35, 37
CLASS WORK/HOME WORK:
• Pages 532 – 533 #’s 4, 8, 10, 12, 16, 22, 24, 34, 36, 38
BELL WORK:
• Solve:
LESSON 7.6
INVERSE TRIGONOMETRIC FUNCTIONS
LET’S REVIEW INVERSES:
• Domain of 𝑓 = Range of 𝑓 −1
• Range of 𝑓 = Domain of 𝑓 −1
• 𝑓 𝑓 −1 𝑥
= 𝑥 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓𝑓 −1
• 𝑓 −1 𝑓 𝑦
= 𝑦 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑦 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓
• The graphs of 𝑓 and 𝑓 −1 are reflections across the line y = x.
• This means the point (a,b) on the graph of 𝑓 is point (b,a) on
the graph of 𝑓 −1 .
INVERSE SINE (𝑠𝑖𝑛−1 )
• The inverse sine function is also referred to as the
arcsine function.
• 𝑦 = 𝑠𝑖𝑛−1 𝑥
𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑦 = arcsin(𝑥)
• Domain: [-1,1]
Range:
𝜋 𝜋
[− , ]
2 2
• What is off about the domain and/or range?
GRAPH OF ARCSINE
• Let’s derive the graph of y = arcsin(x)
USING 𝑠𝑖𝑛−1
• Find the exact value:
• Ex:
1
−1
sin(𝑠𝑖𝑛
)
2
• Ex: 𝑠𝑖𝑛−1 (sin
2𝜋
)
3
INVERSE COSINE (𝑐𝑜𝑠 −1 )
• The inverse cosine function is also referred to as the
arccosine function.
• 𝑦 = 𝑐𝑜𝑠 −1 𝑥
𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑦 = arccos(𝑥)
• Domain: [-1,1]
Range: [0, 𝜋]
GRAPH OF ARCCOSINE
• Let’s derive the graph of y = arccos(x)
USING 𝑐𝑜𝑠 −1
• Find the exact value:
• Ex:
1
−1
cos(𝑐𝑜𝑠
)
2
• Ex: 𝑐𝑜𝑠 −1 (cos
2𝜋
)
3
INVERSE TANGENT (𝑡𝑎𝑛−1 )
• The inverse tangent function is also referred to as
the arctangent function.
• 𝑦 = 𝑡𝑎𝑛−1 𝑥
𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑦 = arctan(𝑥)
• Domain: All Real Numbers
Range:
𝜋 𝜋
(− , )
2 2
GRAPH OF ARCTANGENT
• Let’s derive the graph of y = arctan(x)
USING 𝑡𝑎𝑛−1
• Find the exact value:
• Ex: tan(𝑡𝑎𝑛−1 3)
• Ex: 𝑡𝑎𝑛
−1
2𝜋
(𝑡𝑎𝑛 )
3
HOMEWORK:
• Pages 553 – 554 #’s 1 – 19 odds, 31 and 32
BELL WORK:
• Find the exact value of the following expressions:
3
2
• 1) arccos( )
• 2) arctan(tan
11𝜋
)
4
EXAMPLE:
1
2
4
5
• Find the exact value of sin(arctan − arccos ).
EXAMPLE:
• Find the solutions for the equation on the interval
from [-π,π]. (Round your answers to three decimal
places if necessary.)
• 𝑠𝑖𝑛²𝑥 − 𝑠𝑖𝑛𝑥 − 5 = 0
GRAPHS OF OTHER INVERSES
• **Blue Chart on Page 552**
• These are the graphs for the inverses of cotangent,
secant, and cosecant.
• We are going to skip these!!!
HOMEWORK:
• Pages 553 – 555 #’s 10 – 20 evens, 53, 55, 57
TEST WEDNESDAY
• Lessons 7.1 – 7.6 (no 7.5)
• Proofs
• Solving Trigonometric Equations (For all real numbers
and through intervals)
• Addition and Subtraction Formulas
• Double/Half Angle Identities
• Inverse Functions(Sine, Cosine, and Tangent only)
TEST REVIEW:
• Pages 557 – 559 (Unit 7)
• #’s 1 – 8, 14, 18, 19, 23 – 34, 45, 48, 53, 56
• Pages 620 – 623 (Unit 8)
• #’s 5 – 10, 40 – 45, 47, 48
• As always, these are review problems that are very
similar to problems that you will see on your test!!!
• Also remember to look over previous homework
assignments!!!