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,a≠1.a>0,a≠1.
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(s−a)(s−b)(s−c)A=√s(s−a)(s−b)(s−c)
wheres=a+b+c2s=a+b+c2
Proof:
Leta,a,b,b,andccbe the sides of a triangle, andhhbe the height.

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.

Figure 2.
Using the Pythagorean Theorem,h2+p2=a2h2+p2=a2andh2+q2=b2.h2+q2=b2.
Sinceq=c−p,q=c−p,thenq2=(c−p)2.q2=(c−p)2.Expanding, we find thatq2=c2−2cp+p2.q2=c2−2cp+p2.
We can then addh2h2to each side of the equation to geth2+q2=h2+c2−2cp+p2.
Substitute this result into the equationh2+q2=b2yieldsb2=h2+c2−2cp+p2.
Then replacingh2+p2witha2givesb2=a2−2cp+c2.
Solve forpto get
p=a2+b2−c22c
Sinceh2=a2−p2,we get an expression in terms ofa,b,and c.
h2=a2−p2=(a+p)(a−p)=[a+(a2+c2−b2)2c][a−(a2+c2−b2)2c]=(2ac+a2+c2−b2)(2ac−a2−c2+b2)4c2=((a+c)2−b2)(b2−(a−c)2)4c2=(a+b+c)(a+c−b)(b+a−c)(b−a+c)4c2=(a+b+c)(−a+b+c)(a−b+c)(a+b−c)4c2=2s⋅(2s−a)⋅(2s−b)(2s−c)4c2
Therefore,
h2=4s(s−a)(s−b)(s−c)c2h=2√s(s−a)(s−b)(s−c)c
And sinceA=12ch,then
A=12c2√s(s−a)(s−b)(s−c)c=√s(s−a)(s−b)(s−c)
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+(y−0)2+√(x−c)2+(y−0)2=constant

Figure 3.
Consider a vertex.

Figure 4.
Then,√(x−(−c))2+(y−0)2+√(x−c)2+(y−0)2=2a
Consider a covertex.

Figure 5.
Thenb2+c2=a2.
√(x−(−c))2+(y−0)2+√(x−c)2+(y−0)2=2a√(x+c)2+y2=2a−√(x−c)2+y2(x+c)2+y2=(2a−√(x−c)2+y2)2x2+2cx+c2+y2=4a2−4a√(x−c)2+y2+(x−c)2+y2x2+2cx+c2+y2=4a2−4a√(x−c)2+y2+x2−2cx+y22cx=4a2−4a√(x−c)2+y2−2cx4cx−4a2=4a√(x−c)2+y2−14a(4cx−4a2)=√(x−c)2+y2a−cax=√(x−c)2+y2a2−2xc+c2a2x2=(x−c)2+y2a2−2xc+c2a2x2=x2−2xc+c2+y2a2+c2a2x2=x2+c2+y2a2+c2a2x2=x2+c2+y2a2−c2=x2−c2a2x2+y2a2−c2=x2(1−c2a2)+y2
Let1=a2a2.
Becauseb2+c2=a2,thenb2=a2−c2.
Standard Form of the Hyperbola
1=x2a2−y2b2
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.

Figure 6.
Diagram 1: The difference of the distances from Point P to the foci is constant:
√(x−(−c))2+(y−0)2−√(x−c)2+(y−0)2=constant
Diagram 2: When the point is a vertex, the difference is2a.
√(x−(−c))2+(y−0)2−√(x−c)2+(y−0)2=2a
√(x−(−c))2+(y−0)2−√(x−c)2+(y−0)2=2a√(x+c)2+y2−√(x−c)2+y2=2a√(x+c)2+y2=2a+√(x−c)2+y2(x+c)2+y2=(2a+√(x−c)2+y2)x2+2cx+c2+y2=4a2+4a√(x−c)2+y2x2+2cx+c2+y2=4a2+4a√(x−c)2+y2+x2−2cx+y22cx=4a2+4a√(x−c)2+y2−2cx4cx−4a2=4a√(x−c)2+y2cx−a2=a√(x−c)2+y2(cx−a2)2=a2((x−c)2+y2)c2x2−2a2c2x2+a4=a2x2−2a2c2x2+a2c2+a2y2c2x2+a4=a2x2+a2c2+a2y2a4−a2c2=a2x2−c2x2+a2y2a2(a2−c2)=(a2−c2)x2+a2y2a2(a2−c2)=(c2−a2)x2−a2y2
Definebas a positive number such thatb2=c2−a2.
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(π2−t)sint=cos(π2−t)tant=cot(π2−t)cott=tan(π2−t)sect=csc(π2−t)csct=sec(π2−t) |
Fundamental Identities | tant=sintcostsect=1costcsct=1sintcott=1tant=costsint |
Sum and Difference Identities | cos(α+β)=cosαcosβ−sinαsinβcos(α−β)=cosαcosβ+sinαsinβsin(α+β)=sinαcosβ+cosαsinβsin(α−β)=sinαcosβ−cosαsinβtan(α+β)=tanα+tanβ1−tanαtanβtan(α−β)=tanα−tanβ1+tanαtanβ |
Double-Angle Formulas | sin(2θ)=2sinθcosθcos(2θ)=cos2θ−sin2θcos(2θ)=1−2sin2θcos(2θ)=2cos2θ−1tan(2θ)=2tanθ1−tan2θ |
Half-Angle Formulas | sinα2=±√1−cosα2cosα2=±√1+cosα2tanα2=±√1−cosα1+cosαtanα2=sinα1+cosαtanα2=1−cosαsinα |
Reduction Formulas | sin2θ=1−cos(2θ)2cos2θ=1+cos(2θ)2tan2θ=1−cos(2θ)1+cos(2θ) |
Product-to-Sum Formulas | cosαcosβ=12[cos(α−β)+cos(α+β)]sinαcosβ=12[sin(α+β)+sin(α−β)]sinαsinβ=12[cos(α−β)−cos(α+β)]cosαsinβ=12[sin(α+β)−sin(α−β)] |
Sum-to-Product Formulas | sinα+sinβ=2sin(α+β2)cos(α−β2)sinα−sinβ=2sin(α−β2)cos(α+β2)cosα−cosβ=−2sin(α+β2)sin(α−β2)cosα+cosβ=2cos(α+β2)cos(α−β2) |
Law of Sines | sinαa=sinβb=sinγcasinα=bsinβ=csinγ |
Law of Cosines | a2=b2+c2−2bccosαb2=a2+c2−2accosβc2=a2+b2−2abcosγ |
ToolKit Functions

Figure 7.

Figure 8.

Figure 9.
Trigonometric Functions
Unit Circle

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 | 2√33 | √2 | 2 | Undefined |
Cosecant | Undefined | 2 | √2 | 2√33 | 1 |
Cotangent | Undefined | √3 | 1 | √33 | 0 |
Candela Citations
- Algebra and Trigonometry. Authored by: Jay Abramson, et. al. Provided by: OpenStax CNX. Located at: http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1