Introduction to Trigonometry

Angles

The trigonometric circle
Take an x-axis and an y-axis (orthonormal) and let O be the origin. A circle centered in O and with radius = 1, is called a trigonometric circle or unit circle. Turning counterclockwise is the positive orientation in trigonometry.

Oriented angles
An angle is the figure formed by two rays that have the same beginning point. That point is
called the vertex and the two rays are called the sides of the angle (also legs). If we call [OA the
initial side of the angle and [OB the terminal side, then we have an oriented angle. This angle is referred to as

An oriented angle is in fact the set of all angles which can be transformed to each other
by a rotation and/or a translation.

The introduction of the trigonometric circle makes it possible to attach a value to each oriented angle

trigonometric circle and let the initial side of this angle coincide with the x-axis.
Then the terminal side intersects the trigonometric circle in point Z.

Then Z is the representation of the oriented angle on the trigonometric circle.

There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees.

A circle is divided into 360 equal degrees so that a right angle is 90°. Each degree is subdivided into 60 minutes and each minute into 60 seconds. The symbols °, ‘ and ” are used for degrees, arcminutes, and arcseconds.

In most mathematical work beyond practical geometry, angles are typically measured in radians
rather than degrees.
An angle of 1 radian determines on the circle an arc with length the radius of the circle. Because the length of a full circle is 2πR, a circle contains 2π radians. Contrariwise, if one draws in the center of a circle with radius R an angle of θ radians, then this angle determines an arc on the circle with length θ·R. Subdivisions of radians are written in decimal form. when an angle is represented in radians, one does only mention the value, not the term ‘rad’.

The trigonometric numbers.

Some special angles and their trigonometric numbers

Trigonometric numbers of angles in the other quadrants we shell find through the use of the reference angle (see paragraph 2.6.2.)

Sign variation for the trigonometric numbers by quadrant
Inside a quadrant the trigonometric numbers keep the same sign

sign variation for the trigonometric numbers by quadrant

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:

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 :

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/HypotenuseAdjacent/Hypotenuse = OppositeAdjacent = tan(θ)

So we can say:

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):

Because of all that we can say:

And the other way around:

And we also have:

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:

Related identities include:

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

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)  1cot(B)  cot(A)

Triangle Identities

There are also Triangle Identities that apply to all triangles (not just Right Angled Triangles)

These four formulas convert the product of two cosines and/or sines with a different argument into a sum. The reverse formulas we get by bringing factor ½ to the other side and by substitution: