a, b and c are the sides; f is the angle. This is a Greek letter and is pronounced phi. The angle between sides a and b is 90^o, a right-angle. The side c, which is opposite the right angle, is called the hypotenuse (from a Greek word meaning stretch).

Pythagoras (c582 BC - c497 BC) proved what is now called Pythagoras' Theorem although it had been in use for centuries in the ancient world for building and measuring. The theorem can be described as follows:

This is written mathematically as

If a right-angled triangle has smaller sides of length 3 and 4. What is the length of the hypotenuse?

From Pythagoras' Theorem,

c² = a² + b² = 3² + 4² = (3 × 3) + (4 × 4) = 9 + 16 = 25

If c² = 25, c = 5.

This is the famous 3 : 4 : 5 triangle used in surveying and measuring. There are many such triangles (eg 5 : 12 : 13). You can check that 5 : 12 : 13 is a right-angled triangle by doing the above calculation.

Of course, the numbers do not have to be whole numbers.

A right-angled triangle has a hypotenuse of length 15.3 and one of its other sides is 4.7. Find the legth of the missing side.

From Pythagoras' Theorem,

a² = c² - b² = 15.3² - 4.7² = (15.3 × 15.3) - (4.7 × 4.7) = 234.09 - 22.09 = 212

If a² = 212, a = 14.56.

The area (A) of a right-angled triangle is given by the formula

Find the area of a right-angled triangle with shorter sides of length 4.3 and 6.4 respectively.

The area is given by

A = ab/2 = 4.3 × 6.4 / 2 = 13.76

There are a number of relations between the sides a, b, and c and the angle f. These are called the Trigonometric Functions.

There are three main Trigonometric Functions. These are called Sine, Cosine and Tangent.

This is written as

This is written as

This is written as

The table below shows some of the values of these functions for various angles.

Note the following:

Between 0^o and 90^o:

The values of the Trigonometric Functions (except for 0^o, 30^o, 45^o, 60^o, 90^o) are not whole numbers, fractions or surds. They are transcendental.

The three Trigonometric Functions are related.

This is a side note: The square of a Sine of an angle, say (Sin f)² is more commonly written as Sin²f. This form applies to all the Trigonometric Functions.

Prove that Sin f / Cos f = Tan f

By using the definitions of the Trigonometric Functions

Sin f / Cos f = (a / c) / (b / c) = (a / c) × (c / b) = a / b = Tan f

Prove that Sin²f + Cos²f = 1

By using the definitions of the Trigonometric Functions

Sin²f + Cos²f = (a / c)² + (b / c)² = (a² / c²) + (c² / b²) = (a² + b²) / c².

But a² + b² = c² (from Pythagoras' Theorem)

Therefore (a² + b²) / c² = c² / c² = 1.

Values for the Trigonometric Functions for a particular angle can be found in tables or on a calculator as with Logarithms. We will use them now in some examples.

Find the length of the sides a and c in the following right-angled triangle.

Using the definition of Tangents and rearranging we have

a = b × Tan f = 12.6 × Tan 51^o = 12.6 × 1.235

Using a calculator or tables we can find that Tan 51^o = 1.235 (to three decimal places).

12.6 × 1.235 = 15.56m.

The value of c can be found by using Pythagoras' Theorem. Here we will use the definition of Cosines and rearrange. This gives

c = b / Cos f = 12.6 / Cos 51^o = 12.6 / 0.629 = 20.03m.

Find the angle, f, in the following right-angled triangle.

Using the definition of Tangents

Tan f = a / b = 9.6 / 7.4 = 1.297.

Using tables or a calculator, f = 52.37^o.

The triangle has three sides, a, b, and c. There are three angles, A, B, C (where angle A is opposite side a, etc). The height of the triangle is h.

The sum of the three angles is always 180^o.

The area of this triangle is given by one of the following three formulae

The relationship between the three sides of a general triangle is given by The Cosine Rule. There are three forms of this rule. All are equivalent.

In the first version of the Cosine Rule, if angle A is a right angle, Cos 90^o = 0. The equation then reduces to Pythagoras' Theorem.

a² = b² + c² - (2 × b × c × Cos 90^o) = b² + c² - 0 = b² + c²

The relationship between the sides and angles of a general triangle is given by The Sine Rule.

Find the missing length and the missing angles in the following triangle.

By the Cosine Rule,

a² = b² + c² - (2 × b × c × Cos A)

a² = 6.3² + 4.6² - (2 × 6.3 × 4.6 × Cos 32^o)

a² = 39.69 + 21.16 - (2 × 6.3 × 4.6 × 0.848)

a² = 60.85 - 49.15 = 11.7

a = 3.42m

Now, from the Sine Rule,

a / Sin A = c / Sin C

This can be rearranged to

Sin C = (c × Sin A) / a

By putting in the various values we get

Sin C = (c × Sin A) / a = (4.6 × Sin 32^o) / 3.42 = (4.6 × 0.530) / 3.42 = 0.713

Therefore

C = 45.5^o

The final angle can be found from

A + B + C = 180^o

Rearanging,

B = 180 - A - C = 180 - 32 - 45.5

B = 102.5^o

Using the equations descibed in this essay, it is possible to find out everything about a triangle from just a few given bits of information. In the above example we have calculated that a = 3.42m, B = 102.5^o, C = 45.5^o.

The system of Degrees used for normal angular measurements dates from Babylonian times. A complete circle is 360^o; half a circle is 180^o; and a right angle is 90^o. These numbers were used because they contain many factors and are easy to use. Degrees are artificial units.

When looking at the Trigonometric Functions mathematically, we require a more fundamental unit of angular measure. This is the Radian.

There is a series for evaluating both Sine and Cosine. These series only work if the angle f is in Radians. The series are both valid for all values of f.

Use the series to find the value of Sin 45^o.

Convert the angle to radians:

45^o = p/4 Radians

therefore

Sin 45^o = Sin p/4 = p/4 - ((p/4)³)/3! + ((p/4)⁵)/5! - ....

= 0.785 - 0.081 + 0.002 - ... = 0.706 (to three decimal places).

The correct value is, of course, 0.707.

Look again at the above two series. Now compare them with the Exponential Series below.

With a little bit of mathematics (not here!), it is possible to show that the Trigonometric Functions are related to the number e (2.71828183...), the base of Natural Logarithms) and the Imaginary Number, i.

The relationships are:

 

These equations can be combined and written in an alternative format called Demoivre's Theorem:

We began with right-angled triangles and have ended up with some very abstract equations. Isn't mathematics fun?

What is the value of e^ip?

Using Demoivre's Theorem and remembering that Sin p = 0 and Cos p = -1 (see table above):

e^ip = Cos p + (i × Sin p) = -1 + (i × 0) = -1

These number are discussed further in the Introduction to Numbers web page.