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— Oscar Wilde.

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Trigonometry concept and Formulas

Right Triangle
The **Trigonometric Identities** are equations that are true for Right Angled Triangles. *(If it is not a Right Angled Triangle go to the Triangle Identities page.)*
Each side of a **right triangle** has a name:

**Adjacent** is always next to the angle
And **Opposite** is opposite the angle
We are soon going to be playing with all sorts of functions, but remember it all comes back to that simple triangle with:
Angle θ
Hypotenuse
Adjacent
Opposite
Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent.
They are just the **length of one side divided by another**
For a right triangle with an angle θ :

Sine Function:
**sin(θ) = Opposite / Hypotenuse**
Cosine Function:
**cos(θ) = Adjacent / Hypotenuse**
Tangent Function:
**tan(θ) = Opposite / Adjacent**
For a given angle θ each ratio stays the same
no matter how big or small the triangle is

When we divide Sine by Cosine we get:
*sin(θ)***cos(θ)** = *Opposite/Hypotenuse***Adjacent/Hypotenuse** = *Opposite***Adjacent** = tan(θ)
So we can say:
tan(θ) = *sin(θ)***cos(θ)**
That is our first **Trigonometric Identity**.
Cosecant, Secant and Cotangent
We can also divide “the other way around” (such as **Adjacent/Opposite** instead of **Opposite/Adjacent**):

Cosecant Function:
**csc(θ) = Hypotenuse / Opposite**
Secant Function:
**sec(θ) = Hypotenuse / Adjacent**
Cotangent Function:
**cot(θ) = Adjacent / Opposite**

Example: when Opposite = 2 and Hypotenuse = 4 then
**sin(θ) = 2/4**, and **csc(θ) = 4/2**
Because of all that we can say:
sin(θ) = 1/csc(θ)
cos(θ) = 1/sec(θ)
tan(θ) = 1/cot(θ)

And the other way around:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)

And we also have:
cot(θ) = cos(θ)/sin(θ)

Pythagoras Theorem
For the next trigonometric identities we start with Pythagoras’ Theorem:

The Pythagorean Theorem says that, *in a right triangle,* the square of a plus the square of b is equal to the square of c:
a² + b² = c²
Dividing through by c² gives
a²c² + b²c² = c²c²
This can be simplified to:
(ac)² + (bc)² = 1
Now, **a/c** is **Opposite / Hypotenuse**, which is **sin(θ)**
And **b/c** is **Adjacent / Hypotenuse**, which is **cos(θ)**
So (a/c)² + (b/c)² = 1 can also be written:
sin² θ + cos² θ = 1
Note:**sin² θ** means to find the sine of θ, **then** square the result, and
**sin θ²** means to square θ, **then** do the sine function

Example: 32°
Using **4 decimal places only**:
sin(32°) = 0.5299…
cos(32°) = 0.8480…
Now let’s calculate **sin²θ + cos² θ**:
0.5299² + 0.8480²
= 0.2808… + 0.7191…
= **0.9999…**
We get very close to 1 using only 4 decimal places. Try it on *your* calculator, you might get better results!
Related identities include:
sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ + 1 = sec² θ
tan² θ = sec² θ − 1
cot² θ + 1 = csc² θ
cot² θ = csc² θ − 1
How Do You Remember Them?
The identities mentioned so far can be remembered
using one clever diagram called the Magic Hexagon:

But Wait … There is More!
There are many more identities … here are some of the more useful ones:
Opposite Angle Identities
sin(−θ) = −sin(θ)
cos(−θ) = cos(θ)
tan(−θ) = −tan(θ)
Double Angle Identities

Half Angle Identities
Note that “±” means it may be **either one**, depending on the value of *θ/2*

Angle Sum and Difference Identities
Note that  means you can use plus or minus, and the  means to use the opposite sign.
sin(A  B) = sin(A)cos(B)  cos(A)sin(B)
cos(A  B) = cos(A)cos(B)  sin(A)sin(B)
tan(A  B) = *tan(A)  tan(B)***1  tan(A)tan(B)**
cot(A  B) = *cot(A)cot(B)  1***cot(B)  cot(A)**
Triangle Identities
There are also Triangle Identities which apply to all triangles (not just Right Angled Triangles)

triangle showing Opposite, Adjacent and Hypotenuse — Right Triangle
The **Trigonometric Identities** are equations that are true for Right Angled Triangles. *(If it is not a Right Angled Triangle go to the Triangle Identities page.)*
Each side of a **right triangle** has a name:

**Adjacent** is always next to the angle
And **Opposite** is opposite the angle
We are soon going to be playing with all sorts of functions, but remember it all comes back to that simple triangle with:
Angle θ
Hypotenuse
Adjacent
Opposite
Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent.
They are just the **length of one side divided by another**
For a right triangle with an angle θ :

Sine Function:
**sin(θ) = Opposite / Hypotenuse**
Cosine Function:
**cos(θ) = Adjacent / Hypotenuse**
Tangent Function:
**tan(θ) = Opposite / Adjacent**
For a given angle θ each ratio stays the same
no matter how big or small the triangle is

When we divide Sine by Cosine we get:
*sin(θ)***cos(θ)** = *Opposite/Hypotenuse***Adjacent/Hypotenuse** = *Opposite***Adjacent** = tan(θ)
So we can say:
tan(θ) = *sin(θ)***cos(θ)**
That is our first **Trigonometric Identity**.
Cosecant, Secant and Cotangent
We can also divide “the other way around” (such as **Adjacent/Opposite** instead of **Opposite/Adjacent**):

Cosecant Function:
**csc(θ) = Hypotenuse / Opposite**
Secant Function:
**sec(θ) = Hypotenuse / Adjacent**
Cotangent Function:
**cot(θ) = Adjacent / Opposite**

Example: when Opposite = 2 and Hypotenuse = 4 then
**sin(θ) = 2/4**, and **csc(θ) = 4/2**
Because of all that we can say:
sin(θ) = 1/csc(θ)
cos(θ) = 1/sec(θ)
tan(θ) = 1/cot(θ)

And the other way around:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)

And we also have:
cot(θ) = cos(θ)/sin(θ)

Pythagoras Theorem
For the next trigonometric identities we start with Pythagoras’ Theorem:

The Pythagorean Theorem says that, *in a right triangle,* the square of a plus the square of b is equal to the square of c:
a² + b² = c²
Dividing through by c² gives
a²c² + b²c² = c²c²
This can be simplified to:
(ac)² + (bc)² = 1
Now, **a/c** is **Opposite / Hypotenuse**, which is **sin(θ)**
And **b/c** is **Adjacent / Hypotenuse**, which is **cos(θ)**
So (a/c)² + (b/c)² = 1 can also be written:
sin² θ + cos² θ = 1
Note:**sin² θ** means to find the sine of θ, **then** square the result, and
**sin θ²** means to square θ, **then** do the sine function

Example: 32°
Using **4 decimal places only**:
sin(32°) = 0.5299…
cos(32°) = 0.8480…
Now let’s calculate **sin²θ + cos² θ**:
0.5299² + 0.8480²
= 0.2808… + 0.7191…
= **0.9999…**
We get very close to 1 using only 4 decimal places. Try it on *your* calculator, you might get better results!
Related identities include:
sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ + 1 = sec² θ
tan² θ = sec² θ − 1
cot² θ + 1 = csc² θ
cot² θ = csc² θ − 1
How Do You Remember Them?
The identities mentioned so far can be remembered
using one clever diagram called the Magic Hexagon:

But Wait … There is More!
There are many more identities … here are some of the more useful ones:
Opposite Angle Identities
sin(−θ) = −sin(θ)
cos(−θ) = cos(θ)
tan(−θ) = −tan(θ)
Double Angle Identities

Half Angle Identities
Note that “±” means it may be **either one**, depending on the value of *θ/2*

Angle Sum and Difference Identities
Note that  means you can use plus or minus, and the  means to use the opposite sign.
sin(A  B) = sin(A)cos(B)  cos(A)sin(B)
cos(A  B) = cos(A)cos(B)  sin(A)sin(B)
tan(A  B) = *tan(A)  tan(B)***1  tan(A)tan(B)**
cot(A  B) = *cot(A)cot(B)  1***cot(B)  cot(A)**
Triangle Identities
There are also Triangle Identities which apply to all triangles (not just Right Angled Triangles)

Here you can find a summary of the main formulas you need to know. This list was not organized by years of schooling but thematically. Just choose one of the topics and you will be able to view the formulas related to this subject. This is not an exhaustive list, ie it’s not here all math formulas that are used in mathematics class, only those that were considered most important.

Area

area
Square	A=l2A=l2	ll : length of side
Rectangle	A=w×hA=w×h	ww : width hh : height
Triangle	A=b×h2A=b×h2	bb : base hh : height
Rhombus	A=D×d2A=D×d2	DD : large diagonal dd : small diagonal
Trapezoid	A=B+b2×hA=B+b2×h	BB : large side bb : small side hh: height
Regular polygon	A=P2×aA=P2×a	PP : perimeter aa : apothem
Circle	A=πr2A=πr2 P=2πrP=2πr	rr : radius PP : perimeter
Cone (lateral surface)	A=πr×sA=πr×s	rr : radius ss : slant height
Sphere (surface area)	A=4πr2A=4πr2

Volumes

Cube	V=s3V=s3	ss: side
Parallelepiped	V=l×w×hV=l×w×h	ll: length ww: width hh: height
Regular prism	V=b×hV=b×h	bb: base hh: height
Cylinder	V=πr2×hV=πr2×h	rr: radius hh: height
Cone (or pyramid)	V=13b×hV=13b×h	bb: base hh: height
Sphere	V=43πr3

Functions and equations

Directly Proportional	y=kxy=kx k=yxk=yx	kk: Constant of Proportionality
Inversely Proportional	y=kxy=kx k=yxk=yx
ax2+bx+c=0ax2+bx+c=0	Quadratic formula	x=−b±b2−4ac−−−−−−−√2ax=-b±b2-4ac2a
Concavity	Concave up: a>0a>0
Concave down: a<0a<0
Discriminant	Δ=b2−4acΔ=b2-4ac
Vertex of the parabola	V(−b2a,−Δ4a)V(-b2a,-Δ4a)
y=a(x−h)2+ky=a(x-h)2+k	Concavity	Concave up: a>0a>0
Concave down: a<0a<0
Vertex of the parabola	V(h,k)V(h,k)
Zero-product property	A×B=0⇔A=0∨B=0A×B=0⇔A=0∨B=0	ex : (x+2)×(x−1)=0⇔(x+2)×(x-1)=0⇔ x+2=0∨x−1=0⇔x=−2∨x=1x+2=0∨x-1=0⇔x=-2∨x=1
Difference of two squares	(a−b)(a+b)=a2−b2(a-b)(a+b)=a2-b2	ex : (x−2)(x+2)=x2−22=x2−4(x-2)(x+2)=x2-22=x2-4
Perfect square trinomial	(a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2	ex : (2x+3)2=(2x)2+2⋅2x⋅3+32=(2x+3)2=(2x)2+2⋅2x⋅3+32= 4×2+12x+94×2+12x+9
Binomial theorem (x+y)n=∑k=0n nCk xn−k yk

exponents

Product	am×an=am+nam×an=am+n	ex : 35×32=35+2=3735×32=35+2=37
am×bm=(a×b)mam×bm=(a×b)m	ex : 35×25=(3×2)5=6535×25=(3×2)5=65
Quotient	am÷an=am−nam÷an=am-n	ex : 37÷32=37−2=3537÷32=37-2=35
am÷bm=(a÷b)mam÷bm=(a÷b)m	ex : 65÷25=(6÷2)5=3565÷25=(6÷2)5=35 ex : 53÷23=(52)353÷23=(52)3
Power of Power	(am)p=am×p(am)p=am×p	ex : (52)3=52×3=56(52)3=52×3=56
Zero Exponents	a0=1a0=1	ex : 80=180=1
Negative Exponents	a−n=(1a)na-n=(1a)n	ex : 3−2=(13)23-2=(13)2 ex : (23)−4=(32)4(23)-4=(32)4
Fractional Exponents	apq=ap−−√qapq=apq	ex : 243=24−−√3243=243

Some Quick Rules About Division:

When you divide 0 by another number the answer is always 0. For example: 0 ÷ 2 = 0. That is 0 sweets shared equally among 2 children – each child gets 0 sweets.
When you divide a number by 0 you are not dividing at all (this is quite a problem in mathematics). 2 ÷ 0 is not possible. You have 2 sweets but no children to divide them to. You cannot divide by 0.
When you divide by 1 the answer is the same as the number you were dividing. 2 ÷ 1 = 2. Two sweets divided by one child.
When you divide by 2 you are halving the number. 2 ÷ 2 = 1.
Any number divided by the same number is 1. 20 ÷ 20 = 1. Twenty sweets divided by twenty children – each child gets one sweet.
Numbers must be divided in the correct order. 10 ÷ 2 = 5 whereas 2 ÷ 10 = 0.2. Ten sweets divided by two children is very different to 2 sweets divided by 10 children.
All fractions such as ½, ¼ and ¾ are division sums. ½ is 1 ÷ 2. One sweet divided by two children.

Introduce Yourself (Example Post)

This is an example post, originally published as part of Blogging University. Enroll in one of our ten programs, and start your blog right.

You’re going to publish a post today. Don’t worry about how your blog looks. Don’t worry if you haven’t given it a name yet, or you’re feeling overwhelmed. Just click the “New Post” button, and tell us why you’re here.

Why do this?

Because it gives new readers context. What are you about? Why should they read your blog?
Because it will help you focus you own ideas about your blog and what you’d like to do with it.

The post can be short or long, a personal intro to your life or a bloggy mission statement, a manifesto for the future or a simple outline of your the types of things you hope to publish.

To help you get started, here are a few questions:

Why are you blogging publicly, rather than keeping a personal journal?
What topics do you think you’ll write about?
Who would you love to connect with via your blog?
If you blog successfully throughout the next year, what would you hope to have accomplished?

You’re not locked into any of this; one of the wonderful things about blogs is how they constantly evolve as we learn, grow, and interact with one another — but it’s good to know where and why you started, and articulating your goals may just give you a few other post ideas.

Can’t think how to get started? Just write the first thing that pops into your head. Anne Lamott, author of a book on writing we love, says that you need to give yourself permission to write a “crappy first draft”. Anne makes a great point — just start writing, and worry about editing it later.

When you’re ready to publish, give your post three to five tags that describe your blog’s focus — writing, photography, fiction, parenting, food, cars, movies, sports, whatever. These tags will help others who care about your topics find you in the Reader. Make sure one of the tags is “zerotohero,” so other new bloggers can find you, too.