jeff bivin -- lzhs using fundamental trig identities verifying identities and solving trig equations...
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Jeff Bivin -- LZHS
Using FundamentalTrig Identities
Verifying IdentitiesAnd
Solving Trig Equations
By: Jeffrey BivinLake Zurich High School
Last Updated: December 29, 2009
Jeff Bivin -- LZHS
Reciprocal Identities
1sin
cscu
u
1cos
secu
u
1tan
cotu
u
1csc
sinu
u
1sec
cosu
u
1cot
tanu
u
Jeff Bivin -- LZHS
Quotient Identities
sintan
cos
uu
u
coscot
sin
uu
u
Jeff Bivin -- LZHS
Pythagorean Identities
2 2 1sin cos
2 21 tan sec
2 21 cot csc
Jeff Bivin -- LZHS
Cofunction Identities
2sin cosu u 2cos sinu u
2tan cotu u 2cot tanu u
2sec cscu u 2csc secu u
Complimentary Angles
Jeff Bivin -- LZHS
Even/Odd Identities
sin sinu u cos cosu u
tan tanu u
u u cot cot
sec secu u
csc cscu u
f x f x f x f x
Jeff Bivin -- LZHS
Use the given to evaluate all six trig functions
26265cot sinand
What quadrant is cotangent negative and sine positive??? II
First determine that quadrant the given information holds true…….
tangent
155cot tanif then
2
22626 1cos
226676 1cos 126
2 2526cos
5 2652626
cos
2626 26sin cscif then
526
265
cos
sec
if
then
2 2 1sin cos
Jeff Bivin -- LZHS
Simplify a Trig Expression2sin cos sinx x x
2 1sin cosx x
2x xsin sin
3sin x
2 2 1remember x x : sin cos2 21x x cos sin
Jeff Bivin -- LZHS
Verify a Trig Identity
1
sin coscsc
cos sin
1
1
sin sin cos coscsc
cos sin
2 2
1
sin cos coscsc
cos sin
1
1
coscsc
cos sin
1
cscsin
csc csc
Jeff Bivin -- LZHS
Verify a Trig Identity
1
sin coscsc
cos sin
11
sin cos
cos siny
Use the table feature and graphing utility to check your result.
2 cscy
Select the path style for y2 so you
can see the tracing
Jeff Bivin -- LZHS
Verify a Trig Identitysin cos cos sin
sec cscsin cos
cos sin cos sin cos sinsec csc
sin cos
2 2cos sin cos sin cos sinsec csc
sin cos
1 1sec csc
sin cos
csc sec sec csc
1sec csc
sin cos
Jeff Bivin -- LZHS
Factoring Trig Expressions2 1sin x
1 1sin sinx x
24 3tan tan
4 3 1tan tan
2 1a
1 1a a
24 3a a
4 3 1a a
Jeff Bivin -- LZHS
Factoring Trig Expressions22 6cos cos
2 3 2cos cos
3 1sec
21 1sec sec sec
22 6a a
2 3 2a a
3 1a
21 1a a a
Jeff Bivin -- LZHS
Factoring Trig Expressions2 3sec tan
2 2a a
2 1a a
21 3tan tan
2 2tan tan
2 1tan tan
2 21remember x x : tan sec
Jeff Bivin -- LZHS
Factoring Trig Expressions4 4csc cotx x
2 2 2 2csc cot csc cotx x x x 2 21: cot cscremember x x
2 21 csc cotx x 2 21 csc cotx x
2 2csc cotx x2 21 x x cot cot
21 2 x cot
Jeff Bivin -- LZHS
Factoring Trig Expressions3 2 1cos cos cosx x x
2 1 1 1cos cos cosx x x
21 1cos cosx x
1 1 1cos cos cosx x x
21 1cos cosx x
Jeff Bivin -- LZHS
Add & Subtract Trig Expressions1 1
1 1sin sinx x
1 1 1 1
1 1
sin sin
sin sin
x x
x x
2
1 1
1
sin sin
sin
x x
x
2
2sin
cos
x
x
Jeff Bivin -- LZHS
Add & Subtract Trig Expressions1
1
sin cos
cos sin
x x
x x
2 1 1
1
sin cos cos
cos sin
x x x
x x
2 21 1
1
cos cos
cos sin
x x
x x
22 2
1
cos
cos sin
x
x x
22 1
1
cos
cos sin
x
x x
2 1 1
1
cos cos
cos sin
x x
x x
2 1 cos
sin
x
x
Jeff Bivin -- LZHS
Add & Subtract Trig Expressions2csc
cotcot
xx
x
2 2cot csc
cot
x x
x
2 21csc csc
cot
x x
x
1
cot x
tan x
Jeff Bivin -- LZHS
Verify Trig Identities
Jeff Bivin -- LZHS
Verify Trig Identities - Guidelines
1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.
2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.
3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants with tangents, and cosecants with cotangents.
4. If the preceding guidelines do not help, try converting all terms to sines and cosines.
5. Always try something. Even making an attempt that leads to a dead end provides insight.
Jeff Bivin -- LZHS
Verify Trig Identities2
22
1xx
x
secsin
sec
22
11 x
x sinsec
2 21 x x cos sin
2 2sin sinx x
Jeff Bivin -- LZHS
Verify Trig Identities21 1
21 1
csccos cos
21 1
21 1
cos cos
csccos cos
22
22
1
csc
cos
22
22
x csc
sin
22
12 2
x csc
sin
2 22 2 csc csc
Jeff Bivin -- LZHS
Verify Trig Identitiestan cot sec csc
sin cos
sec csccos sin2 2
sin cossec csc
cos sin1
sec csc
cos sin
1 1
sec csccos sin
sec csc sec csc
Jeff Bivin -- LZHS
Verify Trig Identities
1
cossec tan
sin
1
1 1
cos sin
sec tansin sin
2
1
1
cos sinsec tan
sin
2
1
cos sin
sec tancos
1
sin
sec tancos
1
sin
sec tancos cos
sec tan sec tan
Jeff Bivin -- LZHS
Verify Trig Identities21
1
cos tan
cos sec
x x
x x
21 1
1
x x
x x
cos sec
cos sec
1 11
1
x xx
x x
sec seccos
cos sec
11
xx
x
cossec
cos
1 11
x
x x
cos
cos cos
1 1cos cos
cos cos
x x
x x
Jeff Bivin -- LZHS
Verify Trig Identities 2 22 1sec cotx x
2 2 1x x sec tan
2 21 1x x tan tan
1 1
Jeff Bivin -- LZHS
Verify Trig Identities1 1
1
sin sin
sin cos
x x
x x
1 1
1 1
sin sin
sin sin
x x
x x
2
2
1
1
x
x
sin
sin
22
1 sin
cos
x
x
1 1sin sin
cos cos
x x
x x
Jeff Bivin -- LZHS
Verify Trig Identities3 3
1sin cos
sin cossin cos
x xx x
x x
2 2sin cos sin sin cos cos
sin cos
x x x x x x
x x
2 2sin sin cos cosx x x x
1 1sin cos sin cosx x x x
Jeff Bivin -- LZHS
Verify Trig Identities
2 4 51 2cos sin sin cosx x x x
22 51x x x cos sin cos
22 5x x xcos cos cos
4 5x x xcos cos cos
5 5cos cosx x
2 4 52x x x x x x cos cos sin cos sin cos
Jeff Bivin -- LZHS
Verify Trig Identities4 4 2 41 2 2sin cos cos cosx x x x
2 2 4sin cosx x
2 2 41 cos cosx x
2 4 41 2cos cos cosx x x
2 4 2 41 2 2 1 2 2cos cos cos cosx x x x
Jeff Bivin -- LZHS
Solving Trig Equations
Jeff Bivin -- LZHS
Solving Trig Equations
2 1 0cos x
2 1cos x 12cos x
0 2,on 5
3 3x or x
,on 5
3 32 2x n or x n
Jeff Bivin -- LZHS
Solving Trig Equations2 24 1tan tanx x
23 1tan x 2 1
3tan x
0 2,on 7
6 6x or x
,on 5
6 6x n or x n
33tan x
5 116 6or x or x
Jeff Bivin -- LZHS
Solving Trig Equations2 0sec tanx x
21 0sincos cos
xx x 1 2 0sincos
xx
,on 5
6 62 2x n or x n
1 2 0sincoscos cosx
xx x 1 2 0sin x
12sin x
0cos x 3
2 2,x
Jeff Bivin -- LZHS
Solving Trig Equations0cos cotx x 0cos
sincos xxx
11 0sincos xx
,on
2x n
0cos x 3
2 2,x
0sin x 0,x
11 0sin x 11 sin x1sin x
2x
Jeff Bivin -- LZHS
Solving Trig Equations22 5 3sin sinx x
22 5 3 0sin sinx x
,on 7 116 62 2x n or x n
2 1 3 0sin sinx x
7 116 6x or x
2 1 0sin x 3 0sin x 2 1sin x
12sin x
3sin x
Jeff Bivin -- LZHS
Solving Trig Equations2 2 2 0x cos
2 2 2cos x 222cos x 7
4 42 2 2 2x n or x n 7
8 8x n or x n 0 2,on
9 7 158 8 8 8 , , ,
Jeff Bivin -- LZHS
Sum & Difference Formulas
Jeff Bivin -- LZHS
Sum & Difference Formulas
1tan tantan tantan x y
x yx y
1tan tantan tantan x y
x yx y
sin sin cos cos sinx y x y x y sin sin cos cos sinx y x y x y
cos cos cos sin sinx y x y x y cos cos cos sin sinx y x y x y
Jeff Bivin -- LZHS
S (x3, y3)
P (1, 0)
Q (x1, y1)
R (x2, y2 ) A
B
-B
PR SQ 2 22 22 2 1 3 1 31 0x y x x y y
2 22 22 2 1 3 1 31 0x y x x y y
2 1 3 1 32 2 2x x x y y 2 1 3 1 3x x x y y
cos( ) cos cos( ) sin sin( )A B A B A B cos( ) cos cos sin sinA B A B A B
S (x3, y3) (cos(-B), sin(-B) )Q (x1, y1) (cosA, sinA)
R (x2, y2 ) (cos(A+B), sin(A+B) )
2 2 2 2 2 2
2 2 2 1 1 3 3 1 3 1 32 1 2 2x y x x y x y x x y y
2 2 2 2 2 2
2 2 2 1 1 3 3 1 1 3 32 1 2 2x x y x x x x y y y y
2 1 3 1 31 2 1 1 1 2 2x x x y y
Proof of cos(A+B) = cosA•cosB - sinA•sinB
Jeff Bivin -- LZHS
Proof ofsin(A+B) = sinA•cosB + cosA•sinB
2 2cos cos sin sinA B A B
2sin cosA B A B
2cos A B 2cos A B
A B A B sin cos cos sin
Note: This proof uses the cofunction identities.
Jeff Bivin -- LZHS
Use Sum & Difference Formulas
Find the exact value of sin(750)
75 30 45 30 45sin sin cos cos sino o o o o
75 30 45sin sino o o
32 212 2 2 275sin o
624 475sin o
2 6475sin o
Jeff Bivin -- LZHS
Use Sum & Difference FormulasFind the exact value of:
712 4 3 4 3cos cos cos sin sin
7 3 412 12 12cos cos
32 27 112 2 2 2 2cos
62712 4 4cos
2 6712 4cos
712cos
4 3cos
Jeff Bivin -- LZHS
Use Sum & Difference FormulasFind the exact value of:
3 1
1 3 1
( )
( )
13 13 4 912 12 12 12tan tan tan
1312tan
33 4tan
3 1
1 3
1 3 1 3 1 2 3 3 4 2 31 3 21 3 1 3
2 3
33 4
33 41
tan tan
tan tan
Rationalize the
denominator
Jeff Bivin -- LZHS
1 1 1 1cos sin cos cos sin sin sin cosx x x x
Use Sum & Difference FormulasSimplfy:
2 21 1x x x x
22 1x x
1 1cos sin cosx x
21 x
x1 21 x
x2 21 1x x x x
Jeff Bivin -- LZHS
sin sin cos cos sinu v u v u v
Use Sum & Difference FormulasFind the exact value of sin(u+v) given the following:
5 313 5u and v u v areinQuad IV sin cos , &
3
54
5 313 5sin cos sinu v u v
513
12u v
5 3 12 413 5 13 5sin u v
15 4865 65sin u v
6365sin u v
Jeff Bivin -- LZHS
Verify Trig Identities 2 2sin sin sin sinx y x y x y
sin cos cos sin sin cos cos sinx y x y x y x y
2 2 2 2sin cos cos sinx y x y
2 2 2 21 1sin sin sin sinx y x y
2 2 2 2 2 2sin sin sin sin sin sinx x y y x y
2 2 2 2sin sin sin sinx y x y
Jeff Bivin -- LZHS
Solving Trig Equations on [0, 2π)
4 4 1x x cos cos
74 4x or x
4 4 4 4 1x x x x cos cos sin sin cos cos sin sin
4 4 1x x cos cos cos cos
42 1x cos cos
222 1x cos
2 1x cos2122
x cos
Jeff Bivin -- LZHS
Double-Angle Formulas
Jeff Bivin -- LZHS
Double-Angle Formulas 2x x x sin sin
2x x x cos cos
2x x x x x sin sin cos cos sin
2 2x x xsin sin cos
2x x x x x cos cos cos sin sin
2 22x x x cos cos sin
2 22 1x x x cos sin sin
22 1 2x x cos sin
2 22 1x x x cos cos cos
2 22 1x x x cos cos cos
22 2 1x x cos cos
2 21x x cos sin
2 21x x sin cos
Jeff Bivin -- LZHS
Double-Angle Formulas 2x x x tan tan
12 x xx xx
tan tantan tantan
221
2 xx
x
tantan
tan
Jeff Bivin -- LZHS
Double-Angle Formulas
2 2x x xsin sin cos
2 22x x x cos cos sin
22 1 2x x cos sin
22 2 1x x cos cos
221
2 xx
x
tantan
tan
Jeff Bivin -- LZHS
Half-Angle Formulas
1
2 2
u u
cossin
1
2 2
u u
coscos
1
2 1
u u u
u u
cos sin
tansin cos
2 2
2
u u
u
Note the signs of and
depend on the quadrant in which lies
: sin cos
.
Jeff Bivin -- LZHS
Solve for x
2 0x x sin cos
2 0x x x sin cos cos
2 1 0x x cos sin
0x cos 2 1 0x sin3
2 2x , 2 1x sin12x sin
76 2x n 116 2x n
76 2x n 116 2x n
2x n
2x n
Jeff Bivin -- LZHS
3 4 245 5 25sin 2 2u
sin 2 2sin cosu u u
Use Double-Angle FormulasFind the exact value of sin(2u), cos(2u) & tan(2u)
3
4
5u
35 2sin ,given u u
2cos 2 1 2sinu u 23
5cos 2 1 2u 9
25cos 2 1 2u 18 7
25 25cos 2 1u
221
2 xx
x
tantan
tan
3423
4
2
12x
tan
329161
2x
tan
32716
2xtan
3 16 242 7 72x tan
Jeff Bivin -- LZHS
565
12sin sin2
Use Half-Angle FormulasFind the exact value of: 5
12sin
1
2 2
u u
cossin
561
2
cos
321
2
321
2
2 3
4
2 3
2
Jeff Bivin -- LZHS
747
8tan tan2
Use Half-Angle FormulasFind the exact value of: 7
8tan
74
74
1
cos
sin
1
2 1
u u u
u u
cos sin
tansin cos
2222
1
22
22
1 2
2
2 2 2
2 2
2 2 2
2
2 1 1 2
Jeff Bivin -- LZHS
Solve for x 2 1 0x x sin cos
12 1 0x x cos cos
12 1x x cos cos
2 21
2 1x x cos cos
212 1 2x x x cos cos cos
21 2 4 2x x x cos cos cos20 1 3 2x x cos cos
0 1 2 1x x cos cos
0 1 2 x cos
0 1 x cos
1 2 x cos1 2 x cos12 x cos
3 2n x 53 2n x
1x cos2x n
Jeff Bivin -- LZHS
Product-to-Sum Formulas
12u v u v u v sin sin cos( ) cos( )
12u v u v u v cos cos cos( ) cos( )
12u v u v u v sin cos sin( ) sin( )
12u v u v u v cos sin sin( ) sin( )
Jeff Bivin -- LZHS
Sum-to-Product Formulas
2 22 u v u vu v sin sin sin cos
2 22 u v u vu v sin sin cos sin
2 22 u v u vu v cos cos cos cos
2 22 u v u vu v cos cos sin sin
Jeff Bivin -- LZHS
Power-Reducing Formulas
2 1 2
2
uu
cossin 2 1 2
2
uu
coscos
2 1 2
1 2
uu
u
cos
tancos
Jeff Bivin -- LZHS
24 2sin sinx x
Use Power-Reducing FormulasRewrite sin4x in terms of the first power of cosine.
21 cos22
x
214 1 cos 2x
214 1 2cos 2 cos 2x x
1 cos414 21 2cos 2 xx
18 2 4cos 2 1 cos4x x
18 3 4cos 2 cos 4x x