Trignometry

Introduction to Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

Trigonometry ... is all about triangles.

Right Angled Triangle

The triangle of most interest is the right-angled triangle.

The right angle is shown by the little box in the corner.

We usually know another angle θ.

And we give names to each side:

• Adjacent is adjacent (next to) to the angle θ
• Opposite is opposite the angle θ
• the longest side is the Hypotenuse

A right angled triangle is (you guessed it), a triangle that has a right angle (90°) in it.

The little square in the corner tells us that it is a right angled triangle (I also put 90°, but you don't need to!)

Two Types

There are two types of right angled triangle:

Scalene right angled triangle One right angle Two other unequal angles No equal sides

Isosceles right angled triangle One right angle Two other equal angles always of 45° Two equal sides

Pythagoras' Theorem

Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When the triangle has a right angle (90°) ... ... and squares are made on each of the three sides, then ...
... the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a² + b² = c²

Note:

• c is the longest side of the triangle
• a and b are the other two sides

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a2 + b2 = c2

Example: Solve this triangle.

a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169 c = √169 c = 13

"Sine, Cosine and Tangent"

Trigonometry is good at find a missing side or angle in a triangle.

The special functions Sine, Cosine and Tangent help us!

They are simply one side of a triangle divided by another.

For any angle "θ":

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place): sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57...

Calculators have sin, cos and tan, let's see how to use them:

Example: What is the missing length here?

• We know the Hypotenuse
• We want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

Now multiply by 20 (the Hypotenuse length):

Opposite length = 20 × 0.7071... = 14.14 (to 2 decimals)

Sine, Cosine and Tangent

Three Functions, but same idea.

Right Triangle

Sine, Cosine and Tangent are all based on a Right-Angled Triangle

Before getting stuck into the functions, it helps to give a name to each side of a right triangle: 

• "Opposite" is opposite to the angle θ
• "Adjacent" is adjacent (next to) to the angle θ
• "Hypotenuse" is the long one

Adjacent is always next to the angle And Opposite is opposite the angle

Sine, Cosine and Tangent

Sine, Cosine and Tangent are the three main functions in trigonometry.

They are often shortened to sin, cos and tan.

To calculate them:

Divide the length of one side by another side
... but which sides?

For a triangle with an angle θ, they are calculated this way:

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

In picture form:

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°)	= Opposite / Hypotenuse
	= 2.8 / 4.9
	= 0.57...

How to remember? Think "Sohcahtoa"! It works like this:

Soh...	Sine = Opposite / Hypotenuse
...cah...	Cosine = Adjacent / Hypotenuse
...toa	Tangent = Opposite / Adjacent

You can read more about sohcahtoa ... please remember it, it may help in an exam !

Examples

Example: what are the sine, cosine and tangent of 30° ?

The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √3:

Now we know the lengths, we can calculate the functions:

Sine		sin(30°) = 1 / 2 = 0.5
Cosine		cos(30°) = 1.732 / 2 = 0.866...
Tangent		tan(30°) = 1 / 1.732 = 0.577...

(get your calculator out and check them!)

Example: what are the sine, cosine and tangent of 45° ?

The classic 45° triangle has two sides of 1 and a hypotenuse of √2:

Sine		sin(45°) = 1 / 1.414 = 0.707...
Cosine		cos(45°) = 1 / 1.414 = 0.707...
Tangent		tan(45°) = 1 / 1 = 1

Why?

Why are these functions important?

• Because they let us work out angles when we know sides
• And they let us work out sides when we know angles

Example: Use the sine function to find "d"

We know:

• The cable makes a 39° angle with the seabed
• The cable has a 30 meter length.

And we want to know "d" (the distance down).

Start with:		sin 39° = opposite/hypotenuse
		sin 39° = d/30
Swap Sides:		d/30 = sin 39°
Use a calculator to find sin 39°:		d/30 = 0.6293…
Multiply both sides by 30:		d = 0.6293… x 30
		d = 18.88 to 2 decimal places.

The depth "d" is 18.88 m

Degrees and Radians

Angles can be in Degrees or Radians. Here are some examples:

Angle	Degrees	Radians
Right Angle	90°	π/2
__ Straight Angle	180°	π
Full Rotation	360°	2π

Degrees (Angles)

We can measure Angles in Degrees.

There are 360 degrees in one Full Rotation (one complete circle around). (Angles can also be measured in Radians)

(Note: "Degrees" can also mean Temperature, but here we are talking about Angles)

The Degree Symbol: °

We use a little circle ° following the number to mean degrees.

For example 90° means 90 degrees

One Degree

This is how large 1 Degree is

The Full Circle

A Full Circle is 360° Half a circle is 180° (called a Straight Angle) Quarter of a circle is 90° (called a Right Angle)

Why 360 degrees? Probably because old calendars (such as the Persian Calendar) used 360 days for a year - when they watched the stars they saw them revolve around the North Star one degree per day. Also 360 can be divided evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, which makes a lot of basic geometry easier.

Measuring Degrees

We often measure degrees using a protractor:

The normal protractor measures 0° to 180°

There are also full-circle protractors. But they are not as commonly used because they are a bit big and don't do anything special.

Radians

We can measure Angles in Radians. 1 Radian is about 57.2958 degrees.

Does 57.2958... degrees seem a strange value?
Maybe degrees are strange, as the Radian is a pure measure based on the Radius of the circle:

Radian: the angle made by taking the radius
and wrapping it along the edge of a circle:

So, a Radian "cuts out" a length of a circle's circumference equal to the radius.

Radians and Degrees

So:

• There are π radians in a half circle
• And also 180° in a half circle

So π radians = 180°

So 1 radian = 180°/π = 57.2958° (approximately)

Degrees	Radians (exact)	Radians (approx)
30°	π/6	0.524
45°	π/4	0.785
60°	π/3	1.047
90°	π/2	1.571
180°	π	3.142
270°	3π/2	4.712
360°	2π	6.283

Example: How Many Radians in a Full Circle?

Imagine you cut up pieces of string exactly the length from thecenter of a circle to its edge ...

... how many pieces do you need to go around the edge of the circle?

Answer: 2π (or about 6.283 pieces of string).

Radians Preferred by Mathematicians

Because the radian is based on the pure idea of "the radius being laid along the circumference", it often gives simple and natural results when used in mathematics.

For example, look at the sine function for very small values:

x (radians)	1	0.1	0.01	0.001
sin(x)	0.8414710	0.0998334	0.0099998	0.0009999998

For very small values. "x" and "sin(x)" are almost the same
(as long as "x" is in Radians!)

There will be other examples like that as you learn more about mathematics.

Conclusion

So, degrees are easier to use in everyday work, but radians are much better for mathematics.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

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

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

	The Trigonometric Identities are equations that are true for all right-angled triangles.
	The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle).

Trigonometric Identities

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

Also, when we divide Sine by Cosine we get:

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 ofOpposite/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

a2	+	b2	=	c2
c2		c2		c2

This can be simplified to:

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 sin² θ + 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.

Triangle Identities

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

Magic Hexagon for Trig Identities

This hexagon is a special diagram to help you remember some Trigonometric Identities

Sketch the diagram when you are struggling with trig identities ... it may help you! Here is how:

Building It: The Quotient Identities

Start with: tan(x) = sin(x) / cos(x) To help you remember think "tsc !"
Then add: cot (which is cotangent) on the opposite side of the hexagon to tan csc (which is cosecant) next, and sec (which is secant) last
To help you remember: the "co" functions are all on the right

OK, we have now built our hexagon, what do we get out of it?

Well, we can now follow "around the clock" (either direction) to get all the "Quotient Identities":

Clockwise

tan(x) = sin(x) / cos(x) sin(x) = cos(x) / cot(x) cos(x) = cot(x) / csc(x) cot(x) = csc(x) / sec(x) csc(x) = sec(x) / tan(x) sec(x) = tan(x) / sin(x)

Counterclockwise

cos(x) = sin(x) / tan(x) sin(x) = tan(x) / sec(x) tan(x) = sec(x) / csc(x) sec(x) = csc(x) / cot(x) csc(x) = cot(x) / cos(x) cot(x) = cos(x) / sin(x)

Product Identities

The hexagon also shows that a function between any two functions is equal to them multiplied together (if they are opposite each other, then the "1" is between them):

Example: tan(x)cos(x) = sin(x)	Example: tan(x)cot(x) = 1

Some more examples:

• sin(x)csc(x) = 1
• tan(x)csc(x) = sec(x)
• sin(x)sec(x) = tan(x)

But Wait, There is More!

You can also get the "Reciprocal Identities", by going "through the 1"

Here you can see that sin(x) = 1 / csc(x)

Here is the full set:

• sin(x) = 1 / csc(x)
• cos(x) = 1 / sec(x)
• cot(x) = 1 / tan(x)
• csc(x) = 1 / sin(x)
• sec(x) = 1 / cos(x)
• tan(x) = 1 / cot(x)

Bonus!

AND we also get these:

Examples:

• sin(30°) = cos(60°)
• tan(80°) = cot(10°)
• sec(40°) = csc(50°)

Double Bonus: The Pythagorean Identities

The Unit Circle shows us that

sin² x + cos² x = 1

The magic hexagon can help us remember that, too, by going clockwise around any of these three triangles:

And we have:

• sin²(x) + cos²(x) = 1
• 1 + cot²(x) = csc²(x)
• tan²(x) + 1 = sec²(x)

You can also travel counterclockwise around a triangle, for example:

• 1 - cos²(x) = sin²(x)

Hope this helps you!

Triangle Identities

Triangle Identities

The triangle identities are equations that are true for all triangles (they don't need to have a right angle). For the identities involving right angles triangles see Trigonometric Identities.

Law of Sines

The Law of Sines (also known as The Sine Rule) is:

it can also be this way around:

Law of Cosines

The Law of Cosines (also known as The Cosine Rule) is an extension of the Pythagorean Theorem to any triangle:

which can also be re-arranged to:

Law of Tangents

The Law of Tangents is: