Upload
emery-lyons
View
217
Download
0
Embed Size (px)
Citation preview
(x, y)
(x, - y)(- x, - y)
(- x, y) cosx siny
cosx siny
cosx siny
cos
sintan
cos
sintan
tancos
sin
Sect 5.1 Verifying Trig identities
1sin
csc1
cossec
1tan
cot1
cottan
1sec
cos1
cscsin
Reciprocal
sin 90 cos
cos 90 sin
tan 90 cot
cot 90 tan
sec 90 csc
csc 90 sec
Co-function
sintan
coscos
cotsin
Quotient
2 2
2 2
2 2
sin cos 1
1 tan sec
1 cot csc
Pythagorean
sin sin
cos cos
tan tan
cot cot
sec sec
csc csc
Even/Odd
sec)(a
22
222
yxr
yxr
22 tan1sec
tan
1cot
3
5tan
3
5
cos
sintan
x
y
If and is in quadrant II, find each function value.S AT C
sin)(b cot)(cWhat Trig. Identity has tan and sec?
2
352 1sec
Negative answer.
9252 1sec
9342sec
334sec
Positive answer.
What Trig. Identity has tan and sin?
3453 22 r
34
345
34
5sin
r
y
Positive answer.
What Trig. Identity has tan and cot?
tan
1cot
tan
1cot
35
1cot
5
31
153
35
22 tan1sec
Write cos(x) in terms of tan(x). Secant has a relationship with both tangent and cosine.
1
tan1
cos
1 2
2
22
tan1
1cos
2tan1
1cos
2
2
tan1
tan1cos
Rationalize the denominator.
x2sec
xx 22 csc1cot
xx
2
2
csc1
cot1
Write in terms of sin(x) and cos(x), and simplify the expression so that no quotients appear.
xx 22 cot1csc x2cot x2csc x2csc x2cot
xx
2
2
csc1
cot1
xx
2
2
cot
csc
xxx
2
2
2
sincossin
1
xx
x 2
2
2 cos
sin
sin
1 x2cos
1
Sect 5.2 Verifying Trig identitiesGuidelines to follow.1. Work with one side of the equation at a time. It is often
better to work on the most complicated.
2. Look for opportunities to factor, add fractions, square binomials or multiply a binomial by it’s conjugate to create a monomial.
3. Look to use fundamental identities. Look to see what trig functions are in the answer.
4. Convert everything to sines and cosines
5. Always try something!
Sect 5.2 Verifying Trig identities
Verify.Distribute the cosecant.
Work on the right side first.
sincsccoscsc
Rewrite to sine and cosine.
Simplify the fractions.
Quotient Identity for cotangent.
1cot1cot
sincoscsc1cot
sinsin
1cos
sin
1
1
sin
cos
Sect 5.2 Verifying Trig identities
Verify.Work on the left side first.
Rewrite to sine and cosine.
Simplify the fractions by canceling .
Reciprocal Identity for secant.
xx 22 secsec
xxx 222 seccot1tan
xx
x22
2
sin
1
cos
sin
x2cos
1
Pythagorean Identity
1 + cot2x = csc2x xx 22 csctan
Sect 5.2 Verifying Trig identities
Verify.Work on the left side first.
Rewrite to sine and cosine.
Simplify the fractions by multiplying by the reciprocals and cancel.
Reciprocal Identity for secant and cosecant.
Rewrite the fraction as subtraction of two fractions with the same denominators.
22 cscsec
cossin
cottan
cossin
cot
cossin
tan
cossinsincos
cossincossin
cossin
1
sin
cos
cossin
1
cos
sin
22 sin
1
cos
1
2222 cscseccscsec
Sect 5.2 Verifying Trig identities
Verify.2
22
sec 1sin
sec
Pythagorean Identity
1 + tan2x = sec2x
tan2x = sec2x – 1
Work on the left side first.
2
2
sec
tan
Rewrite to sine and cosine.
2
2
2
cos1
cossin
Rewrite as multiplication.
1
cos
cos
sin 2
2
2
Cancel and Simplify.
22 sinsin
Sect 5.2 Verifying Trig identities
Verify. 2 1 12sec
1 sin 1 sin
Pythagorean Identity
sin2x + cos2x = 1
cos2x = 1 – sin2x
Reciprocal of cosine.
Work on the right side first. Two terms need to be condensed to one term. Find LCD and combine the fractions.
sin1
sin1
sin1
1
sin1
sin1
sin1
1
sin1sin1 LCD
2sin1
22 sin1
sin1
sin1
sin1
2sin1
2
22 cos
12
cos
2
22 sec2sec2
Sect 5.2 Verifying Trig identities
Verify. 2 2 2tan tan 1 cos 1 Work on the right side first. Pythagorean Identities.
sin2x + cos2x = 1
cos2x – 1 = – sin2x
1 + tan2x = sec2x 22 sinsec
Convert to cosine.
22
sincos
1
Multiply.
22 tantan
2
2
cos
sin
Sect 5.2 Verifying Trig identities
Verify. tan cot sec csc
Pythagorean Identity
sin2x + cos2x = 1
Work on the left side first. Try to combine the two terms into one.
sin
cos
cos
sin
Rewrite as two fractions multiplied together.
Reciprocals.
Convert to sine and cosine.
sincos LCD
cos
cos
sin
cos
sin
sin
cos
sin
sincos
cossin 22
sincos
1
sin
1
cos
1
cscseccscsec
Sect 5.2 Verifying Trig identities
Verify. cossec tan
1 sin
Work on the right side first. Two terms need to be condensed to one term.
cos
sin
cos
1 Convert to sine and cosine.
Combine.
cos
sin1When working with binomials, try multiplying by the conjugate to create differences of squares which will incorporate the Pythagorean Identities.
sin1
sin1
cos
sin1
sin1cos
sin1 2
Pythagorean Identity
sin2x + cos2x = 1
cos2x = 1 – sin2x
sin1cos
cos2
Cancel cosine.
sin1
cos
sin1
cos
Sect 5.2 Verifying Trig identities
Verify.2cot 1 sin
1 csc sin
Work on the left side first. Pythagorean Identity and convert to sine and cosine.
1 + cot2x = csc2x
cot2x = csc2x – 1
csc1
1csc2
csc2x – 1 is Diff. of Squares.
Factor.
csc1
1csc1csc
Cancel (csc x + 1)
1csc Convert to sine.
1sin
1
Combine.
sin
sin
sin
1
sin
sin1
sin
sin1
Sect 5.3 Sum and Difference Formulas
AA sin,cos
BB sin,cos
)0,1(
BABA sin,cos
212
212 yyxx
22 0sin1cos BABA
BABABA 22 sin1cos2cos
BA cos22
BABABA sinsin2coscos2cos2
BABABA sinsincoscoscos
A
BA – B
A – B Using Distance Formula
Dist. from (cos(A-B), sin(A-B)) to (1,0) Dist. from (cosA, sinA) to (cosB,sinB)
22 sinsincoscos BABA
BBAABBAA 2222 sinsinsin2sincoscoscos2cos
1 1cos2 BA 1 1 BAcoscos2 BAsinsin2BABA sinsin2coscos22
The Cosine of the Difference of Two Angles
F.O.I.L. F.O.I.L. F.O.I.L.
Pythagorean Identity Pythagorean IdentityPythagorean Identity
Subtract by 2.– 2 – 2
Divide by –2.2 2 2
BABABA sinsincoscoscos
BABABA sinsincoscoscos
BABABA sinsincoscoscos
The Cosine of the Difference of Two Angles
Substitute (-B) for B in the formula to make the Cosine of the Sum of Two Angle.
The Cosine of the Sum of Two Angles
sin (– B) = – sin (B)cos (– B) = cos (B)
90cossin
BABABA sin90sincos90cos90cos
BAsin
To make the Sine of the Sum & Difference of Two Angles we will need the Cofunction Identities for Sine and Cosine.
BA 90cos
90sincos
Start with . BA BABA 90cossin BA 90cos
BABABA sinsincoscoscos
BAsincosBAcossinSubstitute (-B) for B in the formula to make the Sine of the Sum of Two Angle.
BABABA sincoscossinsin
BABABA sincoscossinsin sin (– B) = – sin (B)cos (– B) = cos (B)
BA
BABA
cos
sintan
BABA
BABA
BABA
BABA
BA
coscossinsin
coscoscoscos
coscossincos
coscoscossin
tan
To make the Tangent of the Sum & Difference of Two Angles we will need the Quotient Identities for Tangent.
BABA
BABA
sinsincoscos
sincoscossin
Tricky manipulation: We want this fraction to have tangents in the formula. Need to divide by the same factor in both the top and bottom to make tangents.Start with where we need to divide by cosine.
cos (A)
cos (B)cos (A)
cos (B)
This is what we need divide by all the factors.
BA
BA
tantan1
tantan
BA
BABA
tantan1
tantantan
BABA
BABA
BA
BABA
sinsincoscos
sincoscossin
cos
sintan
BABA
BABA
BABA
BABA
coscossinsin
coscoscoscos
coscossincos
coscoscossin
BA
BA
tantan1
tantan
BA
BABA
tantan1
tantantan
sin (B)sin (A)
This is what we need divide by all the factors.
Find the exact value of .
12
7cos
10515712
1807
12
7
105cosUse the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.
4560cos
BABABA sinsincoscoscos
45sin60sin45cos60cos4560cos45
60
30
1
1
2
3
12
2
2
2
3
2
2
2
1
4
62
4
6
4
2
Find the exact value of .
43
5cos
3006053
1805
3
5
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.
45
60
30
1
1
2
3
1
2
2
2
2
3
2
2
2
1
454
BABABA sinsincoscoscos
45sin300sin45cos300cos45300cos
4
62
4
6
4
2
Suppose that for a Q2 angle and for a Q1 angle .
Find the exact value of each of the following.A. B. C. D.
13
12sin
sinsincoscoscos
5
12 13
5
3
13
12
5
4
13
5
5
3sin
cos cos cos cos
13
5cos
3
4
5
5
4cos
sinsincoscoscos
65
56
65
36
65
20
5
3
13
12
5
4
13
5
65
16
65
36
65
20
Find the exact value of . 75sin
75sinUse the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.
4530sin
BABABA sincoscossinsin
45sin30cos45cos30sin4530sin45
60
30
1
1
2
3
12
2
2
2
3
2
2
2
1
4
62
4
6
4
2
Find the exact value of .
12
7tan
10515712
1807
12
7
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.
4560
30 1
1
2
3
12
13
11
13
1
BA
BABA
tantan1
tantantan
45tan150tan1
45tan150tan45150tan
3
11
13
1
3
1
3
33
3
3
1
13
31
10545150
3
133
31
31
31
Find the exact value of .
12
7tan
10515712
1807
12
7
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.
45
60
301
1
23
1
2
3 1
1 3 1
tan tantan
1 tan tan
A BA B
A B
tan 60 tan 45tan 60 45
1 tan 60 tan 45
1 3
1 3
60 45 105
Find the exact value of . 160sin40cos160cos40sin
120sin 16040sin
BABABA sincoscossinsin
2
360
30
3
1
2
120
Sect 5.5 Dble Angle, Power Reducing, and Half Angle FormulasDouble Angle Formulas: Revise the Sum of Sin, Cos, & Tan Formulas
BABABA sinsincoscoscos
BA
BABA
tantan1
tantantan
AAAAAA sincoscossinsin
BABABA sincoscossinsin
AAA cossin22sin =>
AAAAAA sinsincoscoscos AAA 22 sincos2cos AAA 22 sinsin12cos AA 2sin212cos
AAA 22 cos1cos2cos 1cos22cos 2 AA
AA
AAAA
tantan1
tantantan
A
AA
2tan1
tan22tan
Substitute A in for B.
Find given and . sin 2 ,cos 2 , tan 2 5cos
13
32
2
5
13 1212513 22 y
cossin22sin
13
12sin
169
120
13
5
13
122
5
12tan
22 sincos2cos 169
119
13
12
13
522
2tan1
tan22tan
119
120
512
1
512
2
2
Choose one of the double angle identities to find a value for sine or cosine.
1cos22cos
sin212cos
sincos2cos
2
2
22
Substitute in 4/5.
Subtract by 1.
Divide by -2.
542cos 18090 Find the values of the six trigonometric functions of if and .
2
2
sin215
4
sin212cos
2sin25
1
2sin10
1
sin10
1
Square root both sides, but the answer will be positive, since we are Q2.
10
10
10
1sin
10
3110 22
1
10
103
10
3cos
3
1tan
3cot
3
10sec
10csc
SOH-CAH-TOA
Verify.
Work on the left side first. Convert to sine and cosine with Quotient Identity.
Double angle identity. 2sin(x) cos(x) = sin(2x)
Rewrite the double angle formula.
2cos2x – 1 = cos(2x)
2cos2x = 1 + cos(2x)
Cancel
2cos12sincot
2sinsin
cos
cossin2sin
cos
2cos2
2cos12cos1
2cossincos 22 xxx 72cos7sin7cos 22
x14cos
AAA cossin22sin
22
1
152sin2
1
15cos15sin
4
1
2
1
2
130sin
2
1
Find an identity for cos3 2cos
BABABA sinsincoscoscos
cossin2sin1cos2cos2cos 2 ( )( )
3 2cos 2 2cos cos 2sin cos
3 2cos 2 2cos cos 2 1 cos cos
3 3cos 2 2cos cos 2cos 2cos
34cos 3cos
3cos3 4cos 3cos
Substitute Dble angle Identity.
Pythagorean Identity, rewrite with all cosines.
2sinsin2coscos2cos ( )( )
3 2cos 2 2cos cos 2cos 1 cos
Distribute
BABABA cossinsincoscos
BABABA cossinsincoscos
Product to Sum & Sum to Product FormulasHow to create the Product to Sum Formulas. Add and subtract Sum and Difference formulas for Sine and Cosine.
BABABA cossinsincoscos
BABABA cossinsincoscos
BABABA coscoscoscos2
BABABA coscos2
1coscos
BABABA coscossinsin2
BABABA coscos2
1sinsin
BABABA sinsincoscossin
BABABA sinsincoscossin
BABABA sinsincoscossin
BABABA sinsincoscossin
BABABA sinsincossin2
BABABA sinsin2
1cossin
BABABA sinsinsincos2
BABABA sinsin2
1sincos
BABABA coscoscoscos 21
BABABA coscossinsin 21
BABABA sinsincossin 21
BABABA sinsinsincos 21
Product to Sum Formulas
Sum to Product Formulas
2
cos2
cos2coscosBABA
BA
BABABA coscoscoscos 21
BABABA coscoscoscos2
22cos
22cos
2cos
2cos2
22
yxyxyxyxyxyx
yxBand
yxALet
xyyxyx
coscos2
cos2
cos2
The reason we choose these two fractions for A and B is because we need two values that add up to x and two values that subtract to be y.
BABABA coscoscoscos 21
BABABA coscossinsin 21
BABABA sinsincossin 21
BABABA sinsinsincos 21
Product to Sum Formulas
Sum to Product Formulas
2
cos2
cos2coscosBABA
BA
2
sin2
sin2coscosBABA
BA
2
cos2
sin2sinsinBABA
BA
2
cos2
sin2sinsinBABA
BA
Rewrite as a sum or difference of two functions
Rewrite using sums to product identity.
sin 6 cos 2x x
BABABA sinsincossin 21
xxxxxx 26sin26sin2cos6sin 21
xx 4sin8sin21
xx 3cos4cos
2
sin2
sin2coscosBABA
BA
2
34sin
2
34sin23cos4cos
xxxxxx
2sin
2
7sin2
xx
Half Angle Formulas
2cos 2 1 2sinA A 2cos 2 2cos 1A A
2
2
sintan
2 cos
2Let A
22sin 1 cos 2A A
2 1 cos 2sin
2
AA
1 cossin
2 2
2 1 cos 2cos
2
AA
2 1 cos 2cos
2
AA
2
cos1
2cos
1 cos2
1 cos2
tan2
1 cos2
1 cos2
tan2
cos1
cos1
2tan
The + symbol in each formula DOES NOT mean there are 2 answers, instead it indicates that you must determine the sign of the trigonometric functions based on which quadrant the half angle falls in.
2Let A
cos1
cos1
2tan
cos1
cos1
2tan
cos1cos1
cos1cos1
2
2
2
2
sin
cos1
cos1
cos1
sin
cos1
2tan
cos1cos1
cos1cos1
2
2
2
2
cos1
sin
cos1
cos1
cos1
sin
2tan
Find the exact value for . 5.112cos
Verify the identity.
2sin
2cos1tan
2
225cos5.112cos
2
cos1
2cos
2
225cos1
2
225cos
2
22
1
S A
T C 2
2
222
22
2
22
4
22
cossin2
sin211 2
cossin2
sin2 2
cos
sintan