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Trigonometrytriangle : Pythagoras : angle : sine : cosine : tangent : series : radians : Demoivre |
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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:
This is written mathematically as
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
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
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.
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.
| 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:
Between 0o and 90o:
Finally,
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.
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.
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.
Using the definition of Tangents
Tan φ = a / b = 9.6 / 7.4 = 1.297.
Using tables or a calculator, φ = 52.37o.
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.
The area of this triangle is given by one of the following three formulae
= 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.
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.
Example 9: 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 32o)
a2 = 39.69 + 21.16 - (2 × 6.3 × 4.6 × 0.848)
a2 = 60.85 - 49.15 = 11.7
a = 3.42m
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.
This is written as
This is written as
This is written as
These functions are given here for completeness.
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.
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 |
| 2π | 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.
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
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 Euler's Formula:
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 eiπ?
Using Euler's Formula and remembering that Sin π = 0 and Cos π = -1 (see table above):
eiπ = Cos π + (i × Sin π) = -1 + (i × 0) = -1
These numbers are discussed further in the Introduction to Numbers essay.
© 2000, 2009 KryssTal
Trigonometric Relations
SOS Mathematics lists many of the trigonometric relations.