We now come to an important and fundamental property of the sine and cosine functions. If we know the values of sin a and cos a, and also the
The addition formulas
So far, most of what we have done is to give new names to familiar objects.
But now we will explore the following addition formulas for sines and cosines:
In a sense, they are the key reason why the sine and cosine functions find so many uses in physics, and in mathematics as well.
There are also two related formulas for differences:
Proofs of the addition formulas
The exercises above have shown that the addition and subtraction formulas we propose are reasonable, but if we are to do mathematics, we must have a proof.
We will first prove the addition formula for sin (a+ b) in the case where a, b, and a+ b are all acute angles. We will need two right triangles:
One containing an acute angle equal to a, and another containing an acute angle equal to b.
We must put these triangles together in some way, so that the resulting diagram includes an angle equal to a+ b (we assumed that this angle is also acute). There are only three ways to do this, so that they have a common side:
Each of these pictures gives us a different beautiful proof of the formula for sin (a + b). We explore here the first two. We postpone the third, which is perhaps the most interesting, for another occasion (see the appendix of this chapter).
A first beautiful proof
We start with Fig. 1. Let us label the sides of the triangle as shown. Then sin a = a/c, sin b = e/d. We need to represent sin (a+ b) in the diagram.
Let us draw line D Q perpendicular to the segment marked b:
But DQ, which is related to sin (a+ b), is not related to the ratios representing sin a and sin p. To establish this relationship, we divide D Q into two parts, with a perpendicular from point B:
Now we must relate p/d and a/d to sin a and sin b. We start with the second fraction. The segment a is in triangle A C B, and the segment d is in triangle A B D. We relate the fraction a/d to both triangles by introducing
c as an intermediary (since c is in both triangles):
Putting this all together, we find that
A second beautiful proof
For our second proof, we use the following theorem from Chapter 3 (see page 75).
Theorem The area of a triangle is equal to half the product of two sides and the sine of the angle between them.
In our diagram (see page 126) we have two right triangles, one including an acute angle a and the other including an acute angle b. If we place them so that they have a common side, then we get a new triangle, with one angle equal to a + b:
Ptolemy’s theorem and its connection with the addition formulas
In this appendix we explore the connection between the formula for sin (a + b) and a remarkable geometric theorem of Ptolemy.
The angles of a quadrilateral inscribed in a circle.
Ptolemy’s theorem concerns quadrilaterals that are inscribed in circles. Suppose we have a quadrilateral ABC D, and we want to inscribe it in a circle. This is not always possible. In fact, if there is such a circle, then
We can also show that this condition is sufficient: If the opposite angles of a quadrilateral are supplementary, then the quadrilateral can be inscribed in a circle.
So we have the following results:
Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Example 49 Suppose we want to inscribe a parallelogram in a circle. The result above tell us that its opposite angles must be supplementary, so this parallelogram must be a rectangle. Then the intersection of its diagonals will be the center of the circle, and half the diagonal will be its radius.
The sides of a quadrilateral inscribed in a circle
The theorem of the last section characterizes inscribed quadrilaterals in terms of their angles. Ptolemy’s theorem characterizes them in terms of the length of their sides.
A quadrilateral has four vertices, and so pairs of vertices determine six lengths. Four of these lengths are sides of the quadrilateral, and two of these lengths are diagonals. Ptolemy’s theorem will use these six lengths to tell us whether or not the quadrilateral can be inscribed in a circle.
Ptolemy’s Theorem A quadrilateral can be inscribed in a circle if and only if the product of its diagonals equals the sum of the products of its opposite sides.
Example 50 What does Ptolemy’s theorem tell us for a rectangle? We know that a rectangle can be inscribed in a circle.
That is, Ptolemy’s theorem here reduces to the theorem of Pythagoras. We will not give a geometric proof of Ptolemy’s theorem here. Rather, we will show that it is equivalent to the addition formula for sin(a + b). Ptolemy’s theorem concerns the sides of a quadrilateral. Trigonometry, of course, works with angles. So our first job is to reformulate Ptolemy’s.
Theorem in terms of angles. Let us take a quadrilateral inscribed in a circle of diameter 1.
We know (Chapter 0, page 62) that in such a circle, the length of a chord is equal to the sine of its inscribed angle. If we look at the inscribed angles in the diagram, we find pairs of equal angles. These are labeled with the same Greek letter.
If we have four points A, B, C, and D, then we can divide them into pairs in three different ways:
Each pair of points determines a length. If we take the product of these lengths, then Ptolemy’s theorem says that a circle exists passing through the four points if and only if the sum of two of these products minus the third equals 0. Similarly, if we have n points, a necessary and sufficient condition that they lie on a circle is that the condition of Ptolemy’s theorem is fulfilled for every choice of four of the given points.
Now we can “translate” the lengths of the quadrilateral sides into trigonometric expressions. We have
Now we can write Ptolemy’s theorem in trigonometric form:
This statement is equivalent to the part of Ptolemy’s theorem that says that if a quadrilateral is inscribed in a circle, then the product of the diagonals equals the sum of the products of the opposite sides.
Ptolemy’s theorem is a bit more general than the usual addition formula for sin(a+b), and looks a bit nicer, since it uses only sines, and not cosines.
Ptolemy’s identity implies the addition formula for sines.
which is the usual addition formula. Thus Ptolemy’s theorem implies Ptolemy’s identity, which implies the addition formula for sines.
The addition formulas imply Ptolemy’s theorem Suppose we know the addition formulas for sin(a + b) and cos(a +b).
Let us show that we can use them to prove Ptolemy’s identity.
In Ptolemy’s identity, every term is the product of two sines. In order to derive this identity from the addition formulas, we need to convert these products into sums. The reader is invited to verify, using the formulas for
The right side is a product of two sines. We use (1) to convert this to a sum of cosines:
But this is the same expression that we found equal to the left side. So Ptolemy’s identity follows from the formulas for sine and cosine.