Identities The set of real numbers for which an equation is defined is called the domain of the...
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Transcript of Identities The set of real numbers for which an equation is defined is called the domain of the...
Identities
• The set of real numbers for which an equation is defined is called the domain of the equation. If an equation is true for all values in its domain it is called an identity.
• Example 1. The equation has domain equal to all real numbers
except 1.
• Example 2. The equation has domain equal to all real numbers except those of the form Since it is true for all numbers in its domain, it is an identity.
• Is the equation in Example 1 an identity?
1x1x
1x2
xsecxtan1 22 .n2
Conditional equations
• If an equation is only true for some values in its domain, it is called a conditional equation.
• Example. The equation 2x +1 = 3 has domain equal to all real numbers, but it is only true for x = 1. Therefore, it is a conditional equation.
• Problem. Which equation is conditional?
1 x sin x cos (b)
1x sinx cos a)(
22
22
Fundamental identities
• Reciprocal identities
• Quotient identities
• Pythagorean identities
utan
1 u cot
u cos
1 u sec
u sin
1 u csc
ucot
1u tan
u sec
1u cos
u csc
1 u sin
usin
u cosu cot
u cos
usin u tan
ucscucot 1u secu tan1 1ucos usin222222
Fundamental identities, continued
• Cofunction identities
• Even/Odd identities
u sec u)csc(u cscu)sec(
u tan )u cot(u cot u)(tan
u sinu)( cosu cos usin(
22
22
22 )
ucot u)( cotu sec u)( secu csc u)( csc
utan u)tan(u cosu)( cosu sin u)(sin
Two possible ways to describe a right triangle
x2x4
)arctan(x/22 2
tan2sec2
x0
,t variableindependen isx
20
,t variableindependen is
sec2sec4)tan1(4x4
,tan 2 x
222
Using fundamental identities--an example
• Use fundamental identities to verify the trig identity
• We simplify the left-side of the identity.
sin t.9t)cos(4t)sin(3sin t 2 2π
algebra. usingsin t 9
identity cofunction a usingsin t 4sin t3sin t 2
odd is sine since t)cos(4sin t3sin t 2
t)cos(4t)sin(3sin t 2
2π
2π
Guidelines for verifying trigonometric identities
1. Work with one side of the equation at a time.
2. Look to factor an expression, add fractions, square a binomial, or create a monomial denominator.
3. Look to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.
4. If the preceding guidelines do not help, then try converting all terms to sines and cosines.
5. Always try something.
Practice problems--verify the identity
2
1
2
22
x1
xx) tan(sin6.
sin x xcsc xsec
cot x .5
xcossin x)x)(1sin1( .4
1)1xx)(sec(cot 3.
1 x cos xsec 2.
1 cot x tan x1.
Solving trigonometric equations
• If you are given a trigonometric equation, your task is to manipulate it using algebra and the fundamental trigonometric identities until you can write an equation of the form
• Example. Solve 2∙cos x = 1. Clearly, this is equivalent to cos x = 1/2. To solve for x, note that there are two solutions in [0, 2), namely /3 and 5/3. Also, because cos x has a period of 2, there are infinitely many other solutions, which can be written as
• Since the latter solution gives all possible solutions of the equation, it is called the general solution.
number. function trig
integer.an isn where
nπ23
5π xand nπ2
3
πx
Graphical approaches to solving cos x = 1/2
• The graph indicates how an infinite number of solutions can occur.
• Also, the unit circle shows infinitely many solutions occur.
xcos y
3πx 3
5πx 2πx 3π
2πx 35π 2πx 3
π 4πx 35π
21y
21nπ2
35πcos
21nπ2
3πcos
Solving trigonometric equations--an example
• Solve (tan 3x)(tan x) = tan 3x. This equation can be rewritten as (tan 3x)(tan x – 1) = 0. Next, set the factors to zero. We have
• To solve (i) for 3x in the interval [0, ), we have 3x = 0. In general, we have 3x = n so that x = n/3, n an integer.
• To solve (ii) for x in the interval [0, ), we have x = /4. In general, we have x = /4 + n, n an integer.
1. tan x (ii)
0, 3x tan (i)
Solving trigonometric equations--another example
• Solve csc x + cot x = 1. First, let's convert to sines and cosines. We obtain
• Since there is no obvious way to solve 1 + cos x = sin x directly, we will try squaring both sides in the hope that the Pythagorean identity will result in a simplification. Of course, we know that squaring may introduce extraneous solutions. We have
• After canceling 1, the latter equation becomes
• cos x = 0 yields /2 + 2n, since 3/2 +2n is extraneous, cos x = –1 yields + 2n, which is extraneous since csc x and cot x are undefined at x = .
sin x. x cos1 1 sin x
xcos
xsin
1
x.cos1 xcos x 2cos 1 xsin x)cos1( 2222
0. x)cos (1 xcos 0 x2cos x 2cos 2
Solving trigonometric equations--using inverse functions
• Solve
• The general solution is:
).,(in solutions twoare
)1arctan( x and 3arctan x
01) x )(tan3 x(tan
03tan x 2 x tan
04tan x 2 x tan1
4tan x 2xsec
22
2
2
2
nπ )( xand nπ)3arctan(x 4
Guidelines for solving trigonometric equations
1. Try to isolate the trigonometric function on one side of the equation.
2. Look to use standard techniques such as collecting like terms and factoring (or use the quadratic formula).
3. Look to use the fundamental identities.
4. To solve equations that contain forms such as sin kx or cos kx, first solve for kx and then divide by k.
5. If you can't get a solution using exact values, use inverse trigonometric functions to solve.
Practice problems--solving trigonometric equations
013x tan .7
01 x cosxcos2 .6
0)13x)(sin x (sin 5.
xcos3xsin 4.
sin x 1 x sin3 .3
04xsec3 .2
02 x csc3 .1
2
22
2
0xtan2x tan9.
1xsinxcos .8
2
22
Application of sum and difference formulas
• Sum and Difference Formulas
tan vutan 1
tan vu tan v)tan(u
tan vutan 1
tan vu tan v)tan(u
sin vusin vcosu cos vcos(u
sin vusin vcosu cos vcos(u
sin vu cos vcosusin vsin(u
sin vu cos vcosusin vsin(u
)
)
)
)
Application of sum and difference formulas--two examples
• Find the exact value of
• Verify that
. cos 12π
46 2
22
23
22
21
4π
3π
4π
3π
4π
312π
sin sin cos cos)cos( cos
π
x.cossin x)xsin( 4π2
xcos sin x
) xcossin x(
)sin xcos cos(sin x)xsin(
2
1
2
12
22 4π
4π
4π
Double-angle and power-reducing formulas
• Double-Angle Formulas
• Power-Reducing Formulas (double-angle formulas restated)
u tan1
u tan 2tan(2u)
usin21
1ucos2
usinu cos cos(2u
u cosu2sin sin(2u
2
2
2
22)
)
2u cos1
2u cos1u tan
2
2u cos1ucos
2
2u cos1usin
2
22
Using double-angle formulas to solve an equation
• Solve cos 2x + cos x = 0. First use the double-angle formula for cos 2x.
nπ23
5π x , nπ2
3
πx x cos
2nπ πx1 x cos
2by divide quadratic, factor the 0) x 1)(cos x (cos
xcosunknown with quadratic 01cos(x) x cos2
0cos(x)1) x cos2(
21
21
2
2
n an integer
Using the power-reducing formulas
• Rewrite sin4x in terms of first powers of the cosines of multiple angles.
algebra 4x cos2xcos438
1
reduction-power 2
4x cos12x cos21
4
1
expand x2cos2x cos214
1
reduction-power 2
2x cos1
algebra x)(sin x sin
2
2
224
Other formulas not covered
• Half-Angle formulas, Product-to-Sum Formulas, and Sum-to-Product Formulas are interesting, but will not be covered in this course.
More practice problems--verify using all available identities
xcosxcosxcos 6.
xcos3xcos43x cos 5.
2xsin 1 x)cos(sin x 4.
2x cosxsinxcos 3.
x5cos210x cos1 .2
xcos2
xcsc 2x csc 1.
3π
3π
3
2
44
2
More practice problems--solve for x using all available identities
0 xcos2)xsin( 7.
01x cosπ)cos(x 6.
0xcos)sin(x 5.
0π)sin(x2 π)xtan( .4
1)xsin() sin(x 3.
2 n x)(sin x)(ta x cos 2.
0 x cos 2x sin .1
22π
23π
2
Solution of number 7 from previous slide
• Let u = x/2, then becomes
• Using a double angle formula, we have
• This becomes which factors as
• We have two cases (i) sin u = −1/2, (ii) sin u = 1.
• (i) yields u = 7π/6, u = 11π/6, (ii) yields u = π/2.
• Thus x = 7π/3, x = 11π/3, x = π , which are all the solutions in
[0, 4π).
0 xcos2)xsin( 0.2u cossin(u)
0.u)sin2(1sin(u) 2
0,1usinu sin2 2 0.)1usin1)(usin2(
tan vutan 1
tan vu tan v)tan(u
tan vutan 1
tan vu tan v)tan(u
sin vusin vcosu cos vcos(u
sin vusin vcosu cos vcos(u
sin vu cos vcosusin vsin(u
sin vu cos vcosusin vsin(u
)
)
)
)
formulas difference and Sum
u tan1
u tan 2tan(2u)
usin21
1ucos2
usinu cos cos(2u
u cosu2sin sin(2u
2
2
2
22)
)
formulas angle-Double
2u cos1
2u cos1u tan
2
2u cos1ucos
2
2u cos1usin
2
22
formulas reducing-Power