Proofs, Identities, and Toolkit Functions

Appendix

Important Proofs and Derivations

Product Rule

logaxy=logax+logaylogaxy=logax+logay

Proof:

Letm=logaxm=logaxandn=logay.n=logay.

Write in exponent form.

x=amx=amandy=an.y=an.

Multiply.

xy=aman=am+nxy=aman=am+n

am+n=xyloga(xy)=m+n=logax+logbyam+n=xyloga(xy)=m+n=logax+logby

Change of Base Rule

logab=logcblogcalogab=1logbalogab=logcblogcalogab=1logba

wherexxandyyare positive, anda>0,a1.a>0,a1.

Proof:

Letx=logab.x=logab.

Write in exponent form.

ax=bax=b

Take thelogclogcof both sides.

logcax=logcbxlogca=logcbx=logcblogcalogab=logcblogablogcax=logcbxlogca=logcbx=logcblogcalogab=logcblogab

Whenc=b,c=b,

logab=logbblogba=1logbalogab=logbblogba=1logba

Heron’s Formula

A=s(sa)(sb)(sc)A=s(sa)(sb)(sc)

wheres=a+b+c2s=a+b+c2

Proof:

Leta,a,b,b,andccbe the sides of a triangle, andhhbe the height.

A triangle with sides labeled: a, b and c. A line runs through the center of the triangle, bisecting the top angle; this line is labeled: h.

Figure 1.

Sos=a+b+c2s=a+b+c2.

We can further name the parts of the base in each triangle established by the height such thatp+q=c.p+q=c.

A triangle with sides labeled: a, b, and c. A line runs through the center of the triangle bisecting the angle at the top; this line is labeled: h. The two new line segments on the base of the triangle are labeled: p and q.

Figure 2.

Using the Pythagorean Theorem,h2+p2=a2h2+p2=a2andh2+q2=b2.h2+q2=b2.

Sinceq=cp,q=cp,thenq2=(cp)2.q2=(cp)2.Expanding, we find thatq2=c22cp+p2.q2=c22cp+p2.

We can then addh2h2to each side of the equation to geth2+q2=h2+c22cp+p2.

Substitute this result into the equationh2+q2=b2yieldsb2=h2+c22cp+p2.

Then replacingh2+p2witha2givesb2=a22cp+c2.

Solve forpto get

p=a2+b2c22c

Sinceh2=a2p2,we get an expression in terms ofa,b,and c.

h2=a2p2=(a+p)(ap)=[a+(a2+c2b2)2c][a(a2+c2b2)2c]=(2ac+a2+c2b2)(2aca2c2+b2)4c2=((a+c)2b2)(b2(ac)2)4c2=(a+b+c)(a+cb)(b+ac)(ba+c)4c2=(a+b+c)(a+b+c)(ab+c)(a+bc)4c2=2s(2sa)(2sb)(2sc)4c2

Therefore,

h2=4s(sa)(sb)(sc)c2h=2s(sa)(sb)(sc)c

And sinceA=12ch,then

A=12c2s(sa)(sb)(sc)c=s(sa)(sb)(sc)

Properties of the Dot Product

u·v=v·u

Proof:

u·v=u1,u2,...un·v1,v2,...vn=u1v1+u2v2+...+unvn=v1u1+v2u2+...+vnvn=v1,v2,...vn·u1,u2,...un=v·u

u·(v+w)=u·v+u·w

Proof:

u·(v+w)=u1,u2,...un·(v1,v2,...vn+w1,w2,...wn)=u1,u2,...un·v1+w1,v2+w2,...vn+wn=u1(v1+w1),u2(v2+w2),...un(vn+wn)=u1v1+u1w1,u2v2+u2w2,...unvn+unwn=u1v1,u2v2,...,unvn+u1w1,u2w2,...,unwn=u1,u2,...un·v1,v2,...vn+u1,u2,...un·w1,w2,...wn=u·v+u·w

u·u=|u|2

Proof:

u·u=u1,u2,...un·u1,u2,...un=u1u1+u2u2+...+unun=u12+u22+...+un2=|u1,u2,...un|2=v·u

Standard Form of the Ellipse centered at the Origin

1=x2a2+y2b2

Derivation

An ellipse consists of all the points for which the sum of distances from two foci is constant:

(x(c))2+(y0)2+(xc)2+(y0)2=constant

An ellipse centered at the origin on an x, y-coordinate plane. Points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points appear on the ellipse. Points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points appear on the ellipse. Points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points appear on the x-axis, but not the ellipse. The point (x, y) appears on the ellipse in the first quadrant. Dotted lines extend from F1 and F2 to the point (x, y).

Figure 3.

Consider a vertex.

An ellipse centered at the origin. The points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points are on the ellipse. The points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points are on the ellipse. The points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points are on the x-axis and not on the ellipse. A line extends from the point F1 to a point (x, y) which is at the point (a, 0). A line extends from the point F2 to the point (x, y) as well.

Figure 4.

Then,(x(c))2+(y0)2+(xc)2+(y0)2=2a

Consider a covertex.

An ellipse centered at the origin. The points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points are on the ellipse. The points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points are on the ellipse. The points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points are on the x-axis and not on the ellipse. There is a point (x, y) which is plotted at (0, b). A line extends from the origin to the point (c, 0), this line is labeled: c. A line extends from the origin to the point (x, y), this line is labeled: b. A line extends from the point (c, 0) to the point (x, y); this line is labeled: (1/2)(2a)=a. A dotted line extends from the point (-c, 0) to the point (x, y); this line is labeled: (1/2)(2a)=a.

Figure 5.

Thenb2+c2=a2.

(x(c))2+(y0)2+(xc)2+(y0)2=2a(x+c)2+y2=2a(xc)2+y2(x+c)2+y2=(2a(xc)2+y2)2x2+2cx+c2+y2=4a24a(xc)2+y2+(xc)2+y2x2+2cx+c2+y2=4a24a(xc)2+y2+x22cx+y22cx=4a24a(xc)2+y22cx4cx4a2=4a(xc)2+y214a(4cx4a2)=(xc)2+y2acax=(xc)2+y2a22xc+c2a2x2=(xc)2+y2a22xc+c2a2x2=x22xc+c2+y2a2+c2a2x2=x2+c2+y2a2+c2a2x2=x2+c2+y2a2c2=x2c2a2x2+y2a2c2=x2(1c2a2)+y2

Let1=a2a2.

a2c2=x2(a2c2a2)+y21=x2a2+y2a2c2

Becauseb2+c2=a2,thenb2=a2c2.

1=x2a2+y2a2c21=x2a2+y2b2

Standard Form of the Hyperbola

1=x2a2y2b2

Derivation

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances between two fixed points is constant.

Side-by-side graphs of hyperbole. In Diagram 1: The foci F’ and F are labeled and can be found a little in front of the opening of the hyperbola. A point P at (x,y) on the right curve is labeled. A line extends from the F’ focus to the point P labeled: D1. A line extends from the F focus to the point P labeled: D2. In Diagram 2: The foci F’ and F are labeled and can be found a little in front of the opening of the hyperbola. A point V is labeled at the vertex of the right hyperbola. A line extends from the F’ focus to the point V labeled: D1. A line extends from the F focus to the point V labeled: D2.

Figure 6.

Diagram 1: The difference of the distances from Point P to the foci is constant:

(x(c))2+(y0)2(xc)2+(y0)2=constant

Diagram 2: When the point is a vertex, the difference is2a.

(x(c))2+(y0)2(xc)2+(y0)2=2a

(x(c))2+(y0)2(xc)2+(y0)2=2a(x+c)2+y2(xc)2+y2=2a(x+c)2+y2=2a+(xc)2+y2(x+c)2+y2=(2a+(xc)2+y2)x2+2cx+c2+y2=4a2+4a(xc)2+y2x2+2cx+c2+y2=4a2+4a(xc)2+y2+x22cx+y22cx=4a2+4a(xc)2+y22cx4cx4a2=4a(xc)2+y2cxa2=a(xc)2+y2(cxa2)2=a2((xc)2+y2)c2x22a2c2x2+a4=a2x22a2c2x2+a2c2+a2y2c2x2+a4=a2x2+a2c2+a2y2a4a2c2=a2x2c2x2+a2y2a2(a2c2)=(a2c2)x2+a2y2a2(a2c2)=(c2a2)x2a2y2

Definebas a positive number such thatb2=c2a2.

a2b2=b2x2a2y2a2b2a2b2=b2x2a2b2a2y2a2b21=x2a2y2b2

Trigonometric Identities

Pythagorean Identity cos2t+sin2t=11+tan2t=sec2t1+cot2t=csc2t
Even-Odd Identities cos(t)=costsec(t)=sectsin(t)=sinttan(t)=tantcsc(t)=csctcot(t)=cott
Cofunction Identities cost=sin(π2t)sint=cos(π2t)tant=cot(π2t)cott=tan(π2t)sect=csc(π2t)csct=sec(π2t)
Fundamental Identities tant=sintcostsect=1costcsct=1sintcott=1tant=costsint
Sum and Difference Identities cos(α+β)=cosαcosβsinαsinβcos(αβ)=cosαcosβ+sinαsinβsin(α+β)=sinαcosβ+cosαsinβsin(αβ)=sinαcosβcosαsinβtan(α+β)=tanα+tanβ1tanαtanβtan(αβ)=tanαtanβ1+tanαtanβ
Double-Angle Formulas sin(2θ)=2sinθcosθcos(2θ)=cos2θsin2θcos(2θ)=12sin2θcos(2θ)=2cos2θ1tan(2θ)=2tanθ1tan2θ
Half-Angle Formulas sinα2=±1cosα2cosα2=±1+cosα2tanα2=±1cosα1+cosαtanα2=sinα1+cosαtanα2=1cosαsinα
Reduction Formulas sin2θ=1cos(2θ)2cos2θ=1+cos(2θ)2tan2θ=1cos(2θ)1+cos(2θ)
Product-to-Sum Formulas cosαcosβ=12[cos(αβ)+cos(α+β)]sinαcosβ=12[sin(α+β)+sin(αβ)]sinαsinβ=12[cos(αβ)cos(α+β)]cosαsinβ=12[sin(α+β)sin(αβ)]
Sum-to-Product Formulas sinα+sinβ=2sin(α+β2)cos(αβ2)sinαsinβ=2sin(αβ2)cos(α+β2)cosαcosβ=2sin(α+β2)sin(αβ2)cosα+cosβ=2cos(α+β2)cos(αβ2)
Law of Sines sinαa=sinβb=sinγcasinα=bsinβ=csinγ
Law of Cosines a2=b2+c22bccosαb2=a2+c22accosβc2=a2+b22abcosγ

ToolKit Functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.

Figure 7.

Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.

Figure 8.

Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.

Figure 9.

Trigonometric Functions

Unit Circle

Graph of unit circle with angles in degrees, angles in radians, and points along the circle inscribed.

Figure 10.

Angle 0 π6,or 30° π4,or 45° π3,or 60° π2,or 90°
Cosine 1 32 22 12 0
Sine 0 12 22 32 1
Tangent 0 33 1 3 Undefined
Secant 1 233 2 2 Undefined
Cosecant Undefined 2 2 233 1
Cotangent Undefined 3 1 33 0