Trigonometry

Trigonometry

triangle : Pythagoras : angle : sine : cosine : tangent : series : radians : Demoivre


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Right Angled Triangles

Consider the right-angled triangle shown in the diagram.

Right-Angled Triangle

a, b and c are the sides; φ is the angle. This is a Greek letter and is pronounced phi. The angle between sides a and b is 90o, 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:

In a right-angles triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

This is written mathematically as

a2 + b2 = c2

Example 1: 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.

Example 2: A right-angled triangle has a hypotenuse of length 15.3 and one of its other sides is 4.7. Find the length 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

A = ab/2

Example 3: 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


The Trigonometric Functions

Let us return to the diagram of the right-angled triangle.

Right-Angled Triangle

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

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

The Sine of the angle φ is defined as the length of the opposite side (opposite to the angle φ) divided by the hypotenuse.

This is written as

Sin φ = a / c

The Cosine of the angle φ is defined as the length of the adjacent side (adjacent to the angle φ) divided by the hypotenuse.

This is written as

Cos φ = b / c

The Tangent of the angle φ is defined as the length of the opposite side (opposite to the angle φ) divided by the length of the adjacent side (adjacent to the angle φ).

This is written as

Tan φ = a / b

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

Angle Sin Cos Tan
0o 0.000 1.000 0.000
30o 0.500 0.866 0.577
45o 0.707 0.707 1.000
60o 0.866 0.500 1.732
90o 1.000 0.000 Infinite

Note the following:

Sin 0o = Cos 90o = 0

Sin 30o = Cos 60o = 0.500

Sin 45o = Cos 45o = 0.707 = 1 / (√2)

Sin 60o = Cos 30o = 0.866 = (√3) / 2

Sin 90o = Cos 0o = 1

Between 0o and 90o:

Sines increase from 0 to 1,
Cosines decrease from 1 to 0,
Tangents increase from 0 to infinity.

Finally,
Cos(90 - X) = Sin(X)
Sin(90 - X) = Cos(X)

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


The three Trigonometric Functions are related.

Sin φ / Cos φ = Tan φ

Sin2φ + Cos2φ = 1

Note: The square of a Sine of an angle, say (Sin φ)2 is more commonly written as Sin2φ. This form applies to all the Trigonometric Functions.


Example 4: Prove that Sin φ / Cos φ = Tan φ

By using the definitions of the Trigonometric Functions

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

Example 5: Prove that Sin2φ + Cos2φ = 1

By using the definitions of the Trigonometric Functions

Sin2φ + Cos2φ = (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.

Example 6: Find the length of the sides a and c in the following right-angled triangle.

Triangle

Using the definition of Tangents and rearranging we have

a = b × Tan φ = 12.6 × Tan 51o = 12.6 × 1.235

Using a calculator or tables we can find that Tan 51o = 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 φ = 12.6 / Cos 51o = 12.6 / 0.629 = 20.03m.

Example 7: Find the angle, φ, in the following right-angled triangle.

Triangle

Using the definition of Tangents

Tan φ = a / b = 9.6 / 7.4 = 1.297.

Using tables or a calculator, φ = 52.37o.


The Sine and Cosine Rules

So far, we have been looking at right-angle triangles. In general, triangles can have any angles. Consider the triangle below.

General Triangle

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 180o.

A + B + C = 180o

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

Area = (a × b × Sin C) / 2 = (a × c × Sin B) / 2 = (b × c × Sin A) / 2

= b × h / 2

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.

a2 = b2 + c2 - (2 × b × c × Cos A)

b2 = a2 + c2 - (2 × a × c × Cos B)

c2 = a2 + b2 - (2 × a × b × Cos C)

Example 8: Show that Pythagoras' Theorem is a special case of the Cosine Rule.

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

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

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

a / Sin A = b / Sin B = c / Sin C

Example 9: Find the missing length and the missing angles in the following triangle.

General Triangle

By the Cosine Rule,

a2 = b2 + c2 - (2 × b × c × Cos A)

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

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 32o) / 3.42 = (4.6 × 0.530) / 3.42 = 0.713

Therefore

C = 45.5o

The final angle can be found from

A + B + C = 180o

Rearanging,

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

B = 102.5o

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 = 102.5o, C = 45.5o.


More Trigonometric Functions

Apart from the three trigonometric functions already defined, there are three more which are their reciprocals.

The Secant of the angle φ is defined as the reciprocal of the Cosine.

This is written as

Sec φ = 1 / Cos φ

The Cosecant of the angle φ is defined as the reciprocal of the Sine.

This is written as

Csc φ = 1 / Sin φ

The Cotangent of the angle φ is defined as the reciprocal of the Tangent.

This is written as

Cot φ = 1 / Tan φ

These functions are given here for completeness.


Trigonometric Series

Before we look at series for the Trigonometric Functions, I want to talk about angles.

The system of Degrees used for normal angular measurements dates from Babylonian times. A complete circle is 360o; half a circle is 180o; and a right angle is 90o. 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.

The Radian is defined so that a complete circle is 2π Radians.
A half circle is π Radians and a right angle is π/2 Radians.

1 Radian = 57.3o

1o = 0.0175 Radians

Radians Degrees Sin Cos Tan
0 0 0 1 0
π/2 90 1 0 Infinite
π 180 0 -1 0
3π/2 270 -1 0 Infinite
360 0 1 0


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

Sine Series

Cosine Series

Example 10: Use the series to find the value of Sin 45o.

Convert the angle to radians:

45o = π/4 Radians

therefore

Sin 45o = Sin π/4 = π/4 - ((π/4)3)/3! + ((π/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.

Exp Series

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:

Sine   Cosine

These equations can be combined and written in an alternative format called Euler's Formula:

e = Cos φ + i × Sin φ

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

Example 11: What is the value of e?

Using Euler's Formula and remembering that Sin π = 0 and Cos π = -1 (see table above):

e = Cos π + (i × Sin π) = -1 + (i × 0) = -1

These numbers are discussed further in the Introduction to Numbers essay.

© 2000, 2009 KryssTal


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KryssTal Related Pages

Introduction to different types of numbers: Real, Imaginary, Rational, Irrational, Transcendental.

An introduction to algebra and how to solve simple, simultaneous and Quadratic equations.

A series devised by Isaac Newton that is used for calculations. More on indices: roots and powers. Factorials. Combinations.

Index and base. Logarithms defined. Base 10 and base e. Uses of logarithms in calculations. Series for logarithms.

How to solve equations containing sines, cosines and tangents.

Spherical Trigonometry is the trigonometry of triangles drawn on a sphere.


External Trigonometric Link

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Trigonometric Relations
SOS Mathematics lists many of the trigonometric relations.