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