[PDF] Trigonometric Formula Sheet - Definition of the Trig Functions

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Trigonometric Formula Sheet

Denition of the Trig Functions

Right Triangle Denition

Assume that:

0< <2

or 0< <90hypotenuse adjacentopposite sin=opphyp csc=hypopp cos=adjhyp sec=hypadj tan=oppadj cot=adjoppUnit Circle Denition

Assumecan be any angle.xy

y x1(x;y) sin=y1 csc=1y cos=x1 sec=1x tan=yx cot=xy

Domains of the Trig Functions

sin;82(1;1) cos;82(1;1) tan;86= n +12 ; where n2Zcsc;86=n; where n2Z sec;86= n+12 ; where n2Z cot;86=n; where n2Z

Ranges of the Trig Functions

1sin1 1cos1

1 tan 1csc1andcsc 1

sec1andsec 1

1 cot 1

Periods of the Trig Functions

The period of a function is the number, T, such that f (+T ) = f () . So, if!is a xed number andis any angle we have the following periods. sin(!))T=2! cos(!))T=2! tan(!))T=! csc(!))T=2! sec(!))T=2! cot(!))T=! 1

Identities and Formulas

Tangent and Cotangent Identities

tan=sincoscot=cossin

Reciprocal Identities

sin=1csccsc=1sin cos=1secsec=1cos tan=1cotcot=1tan

Pythagorean Identities

sin

2+ cos2= 1

tan

2+ 1 = sec2

1 + cot

2= csc2

Even and Odd Formulas

sin() =sin cos() = cos tan() =tancsc() =csc sec() = sec cot() =cot

Periodic Formulas

If n is an integer

sin(+ 2n) = sin cos(+ 2n) = cos tan(+n) = tancsc(+ 2n) = csc sec(+ 2n) = sec cot(+n) = cot

Double Angle Formulas

sin(2) = 2sincos cos(2) = cos2sin2 = 2cos 21
= 12sin2 tan(2) =2tan1tan2

Degrees to Radians Formulas

Ifxis an angle in degrees andtis an angle in

radians then: 180
=tx )t=x180 andx=180tHalf Angle Formulas sin=r1cos(2)2 cos=r1 + cos(2)2 tan=s1cos(2)1 + cos(2)

Sum and Dierence Formulas

sin() = sincoscossin cos() = coscossinsin tan() =tantan1tantan

Product to Sum Formulas

sinsin=12 [cos()cos(+)] coscos=12 [cos() + cos(+)] sincos=12 [sin(+) + sin()] cossin=12 [sin(+)sin()]

Sum to Product Formulas

sin+ sin= 2sin+2 cos2 sinsin= 2cos+2 sin2 cos+ cos= 2cos+2 cos2 coscos=2sin+2 sin2

Cofunction Formulas

sin 2 = cos csc 2 = sec tan 2 = cotcos 2 = sin sec 2 = csc cot 2 = tan 2

Unit Circle

0 ;2(1;0)180 ;(1;0)(0;1)90 ;2 (0;1)270 ;3230 ;6( p3 2 ;12 )45 ;4( p2 2 ;p2 2 )60 ;3( 12 ;p3 2 )120 ;23(12 ;p3 2 )135 ;34(p2 2 ;p2 2 )150 ;56(p3 2 ;12 )210 ;76 (p3 2 ;12 )225 ;54 (p2 2 ;p2 2 )240 ;43 (12 ;p3 2 )300 ;53 12 ;p3 2 )315 ;74 p2 2 ;p2 2 )330 ;116 p3 2 ;12 )For any ordered pair on the unit circle(x;y) : cos=x andsin=y

Example

cos( 76
) =p3 2 sin(76 ) =12 3

Inverse Trig Functions

Denition

= sin1(x)is equivalent tox= sin = cos1(x)is equivalent tox= cos = tan1(x)is equivalent tox= tan

Domain and Range

Function

= sin1(x) = cos1(x) = tan1(x)Domain 1x1 1x1

1 x 1Range

2 2 0 2 < <2

Inverse Properties

These properties hold for x in the domain andin

the range sin(sin

1(x)) =x

cos(cos

1(x)) =x

tan(tan

1(x)) =xsin

1(sin()) =

cos

1(cos()) =

tan

1(tan()) =

Other Notations

sin

1(x) = arcsin(x)

cos

1(x) = arccos(x)

tan

1(x) = arctan(x)

Law of Sines, Cosines, and Tangentsa

bc

Law of Sines

sina =sinb =sin c

Law of Cosines

a

2=b2+c22bccos

b

2=a2+c22accos

c

2=a2+b22abcos

Law of Tangents

aba+b=tan12 ()tan 12 bcb+c=tan12 )tan 12 aca+c=tan12 )tan 12 4

Complex Numbers

i=p1i2=1i3=i i4= 1 pa=ipa; a0 (a+bi) + (c+di) =a+c+ (b+d)i (a+bi)(c+di) =ac+ (bd)i (a+bi)(c+di) =acbd+ (ad+bc)i(a+bi)(abi) =a2+b2 ja+bij=pa

2+b2Complex Modulus(a+bi) =abiComplex Conjugate(a+bi)(a+bi) =ja+bij2

DeMoivre's Theorem

Letz=r(cos+isin), and letnbe a positive integer.

Then: z n=rn(cosn+isinn):

Example:Letz= 1i, ndz6.

Solution: First writezin polar form.

r=p(1)

2+ (1)2=p2

=arg(z) = tan111 =4

Polar Form:z=p2

cos 4 +isin 4

Applying DeMoivre's Theorem gives :

z 6=p2 6 cos 6 4 +isin 6 4 = 2 3 cos 32
+isin 32
= 8(0 +i(1)) = 8i 5 Finding thenthroots of a number using DeMoivre's Theorem Example:Find all the complex fourth roots of 4. That is, nd all the complex solutions of x 4= 4.

We are asked to nd all complex fourth roots of 4.

These are all the solutions (including the complex values) of the equationx4= 4. For any positive integern, a nonzero complex numberzhas exactlyndistinctnth roots. More specically, ifzis written in the trigonometric formr(cos+isin), thenth roots of zare given by the following formula. ()r1n cosn +360kn
+isinn +360kn
; for k= 0;1;2;:::;n1: Remember from the previous example we need to write 4 in trigonometric form by using: r=p(a)2+ (b)2and=arg(z) = tan1ba

So we have the complex numbera+ib= 4 +i0.

Thereforea= 4 andb= 0

Sor=p(4)

2+ (0)2= 4 and

=arg(z) = tan104 = 0

Finally our trigonometric form is 4 = 4(cos0

+isin0) Using the formula () above withn= 4, we can nd the fourth roots of 4(cos0+isin0)

Fork= 0;414

cos04 +36004
+isin04 +36004
=p2(cos(0 ) +isin(0)) =p2

Fork= 1;414

cos04 +36014
+isin04 +36014
=p2(cos(90 ) +isin(90)) =p2i

Fork= 2;414

cos04 +36024
+isin04 +36024
=p2(cos(180 ) +isin(180)) =p2

Fork= 3;414

cos04 +36034
+isin04 +36034
=p2(cos(270 ) +isin(270)) =p2i

Thus all of the complex roots ofx4= 4 are:

p2;p2i;p2;p2i. 6

Formulas for the Conic Sections

Circle

StandardForm: (xh)2+ (yk)2=r2

Where(h;k) =centerandr=radius

Ellipse

Standard Form for Horizontal Major Axis:

(xh)2a

2+(yk)2b

2=1

Standard Form for V ertical Major Axis:

(xh)2b

2+(yk)2a

2=1

Where (h;k)= center

2a=length of major axis

2b=length of minor axis

(0Foci can be found by usingc2=a2b2

Wherec= foci length

7

More Conic Sections

Hyperbola

Standard Form for Horizontal Transverse Axis:

(xh)2a

2(yk)2b

2=1

Standard Form for V ertical Transverse Axis:

(yk)2a

2(xh)2b

2=1

Where (h;k)= center

a=distance between center and either vertex

Foci can be found by usingb2=c2a2

Wherecis the distance between

center and either focus. (b>0)

Parabola

Vertical axis:y=a(xh)2+k

Horizontal axis:x=a(yk)2+h

Where (h;k)= vertex

a=scaling factor 8 x

Example: sin(

(5π4 )=p2

2f(x)f(x) = sin(x)0π

6π 4π 3π -11

2⎷2

2⎷3

2 12 ⎷2 2 ⎷3 2x

Example: cos(

(7π6 )=p3

2f(x)f(x) = cos(x)0π

6π 4π 3π -11

2⎷2

2⎷3

2 12 ⎷2 2 ⎷3 2 9 xf(x)f(x) = tanxπ 2- π2 ⎷3

31⎷3

⎷3 3 -1- ⎷3π 4-

π40π

6-

π6π

3-

π32π3-

2π33π4-

3π45π6-

5π6π-π10

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