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Construction of Right Triangles
Right triangles are easily constructed. Recall that a diameter is a straight line which starts at one point on the circle and goes through the center to the other side. Using this property of a circle:
- Construct a diameter of a circle. Call the points where the diameter touches the circle a and c.
- Choose a distinct point b on the circle (any point that is neither a nor c).
- Construct line segments connecting points a, b and c.
Provided the above three directions are followed, the resulting triangle Δabc will be a right triangle. This result is known as Thales' theorem.
This right triangle can be further divided into two isosceles triangles by adding a line segment from b to the center of the circle.
To simplify the following discussion, we specify that the circle has diameter 1 and is oriented such that the diameter drawn above runs left to right. We shall denote the angle of the triangle at the right θ and that at the left φ. The sides of the right triangle will be labeled, starting on the right, a, b, c, so that a is the rightmost side, c the leftmost, and b the diameter. We know from earlier that side b is opposite the right angle and is called the hypotenuse. Side a is opposite angle φ, while side c is opposite angle θ. We reiterate that the diameter, now called b, shall be assumed to have length 1 except where otherwise noted.
By constructing a right angle to the diameter at one of the points where it crosses the circle and then using the method outlined earlier for producing binary fractions of the right angle, we can construct one of the angles, say θ, as an angle of known measure between 0 and π / 2. The measure of the other angle φ is then π - π/2 - θ = π/2 - θ, the complement of angle θ. Likewise, we can bisect the diameter of the circle to produce lengths which are binary fractions of the length of the diameter. Using a compass, a binary fractional length of the diameter can be used to construct side a (or c) having a known size (with regard to the diameter b) from which side c can be constructed.
Using Right Triangles to find Unknown Sides
Finding unknown sides from two other sides
From the Pythagorean Theorem, we know that:
Recall that c (the diameter/hypotenuse in the above example) has been defined as having a length of one, therefore:
permits us to calculate the length of b squared. It may happen that b squared is a fraction such as 1/4 for which a rational square root can be found, in this case as 1/2, alternatively, we could use Newton's Method to find an approximate value for b.
Finding unknown sides from one side and one (non-right) angle
As any triangle could be compared to our basic triangle (formed from a circle with a diameter of one), a table enumerating the relationships between angles and side lengths would be very useful to understanding the properties of any triangle. However, such a table would be unwieldy in practice and it is often not necessary to know the exact value.
Of course, given an angle , we could construct a right angled triangle using ruler and compass that had as one of its angles, we could then measure the length of the side that corresponds to a to evaluate the function. Such measurements would necessarily be inexact; it would be a problem in physics to see how accurately such measurements can be made; using trigonometry we can make precise predictions with which the results of these physical measurements can be compared.
The most common way of communicating the idea that relationships exist without providing exact details takes the form of a 'function'. A function is like a machine that takes some simple input and produces some simple output. Usually a function defines some kind of rule ([Function]) and provides us a handy notation useful in trigonometry. That way, we know that we're using a certain relationship without needing to know the exact numerical values. Basic trigonometric functions are simply stand-ins for the relationship between angles and sides of a triangle.
One such function, which allows us to know the relationship between any value of and the corresponding value of is called the cosine or . This universal relationship is represented as: which would save us the work of constructing angles and lengths and making difficult deductions from them.
This means that if you know the cosine of an angle, you also know the relationship between the lengths of the sides. The actual size of the triangle can be bigger or smaller, but the mathematical relationship represented by the cosine does not change so long as the size of the angle remains constant.
Some explicit values for the cos function are known. For , sides and coincide: , so . For , sides and coincide and are of length 1, and side is of zero length, consequently , . and .
The simplest right angle triangle we can draw is the isosceles right angled triangle, it has a pair of angles of size radians, and if its hypotenuse is considered to be of length one, then the sides and are of length as can be verified by the theorem of Pythagoras. If the side is chosen to be the same length as the radius of the circle containing the right angled triangle, then the right hand isosceles triangle obtained by splitting the right angled triangle from the circumference of the circle to its center is an equilateral triangle, so must be , and must be and must be .
Properties of the cosine and sine functions
A full revolution is an angle of 2π radians, so increasing an angle by this amount gets you exactly back to your starting point. Therefore, a perfect circle is maintained, along with the relationships formed by triangles within, by adding 2π to any angle θ. This is called the period, --the size of the angle or the time period over which the relationships begin to repeat (correlating the two is complex, and allows us to talk about wave theory).
Using functions, we can represent this fact in terms of the cosine function by stating that :
Knowing the period of the sine and cosine functions (and by derivation, that of other functions) is useful because it means that we can substitute one angle for another when we know that the period is the same. This helps in calculations, such as when there is the need to add or subtract angles.
Half Angle and Double Angle Formulas
We can derive a formula for cos(θ/2) in terms of cos(θ) which allows us to find the value of the cos() function for many more angles. To derive the formula, draw an isosceles triangle, draw a circle through its corners, connect the center of the circle with radii to each corner of the isoceles triangle, extend the radius through the apex of the isoceles triangle into a diameter of the circle and connect the point where the diameter crosses the other side of the circle with lines to the other corners of the isoceles triangle.
which gives a method of calculating the cos() of half an angle in terms of the cos() of the original angle. For this reason * is called the "Cosine Half Angle Formula".
The half angle formula can be applied to split the newly discovered angle which in turn can be split again ad infinitum. Of course, each new split involves finding the square root of a term with a square root, so this cannot be recommended as an effective procedure for computing values of the cos() function.
Equation * can be inverted to find cos(θ) in terms of cos(θ)/2:
that is a formula for the cos() of double an angle in terms of the cos() of the original angle, and is called the "cosine double angle sum formula".