What is new about trigonometry?
Two of the most basic figures studied in geometry are the triangle and the circle. Trigonometry will tell us more than we learned in geometry abouteach of these figures.
For example, in geometry we learn that if we know the lengths of the three sides of a triangle, then the measures of its angles are completely determined1 (and, in fact, almost everything else about the triangle is determined).
But, except for a few very special triangles, geometry does not tell us how to compute the measures of the angles, given the measures of the sides.
Example 1 The measures of the sides of a triangle are 6, 6, and 6 centimeters.
What are the measures of its angles?
It is sometimes said that the lengths of three sides determine a triangle, but one must be careful in thinking this way. Given three arbitrary lengths, one may or may not be able to form a triangle (they form a triangle if and only if the sum of any two of them is greater than the third). But if one can form a triangle, then the angles of that triangle are indeed determined.
Solution. The triangle has three equal sides, so its three angles are also equal. Since the sum of the angles is 180″, the degree-measure of each angle is 180/3 = 60°. Geometry allows us to know this without actually measuring the angles, or even drawing the triangle.
Example 2 The measures of the sides of a triangle are 5, 6, and 7 centimeters.
What are the measures of its angles?
Solution. We cannot find these angle measures using geometry.
The best we can do is to draw the triangle, and measure the angles with a protractor.
But how will we know how accurately we have measured? We willanswer this question in Chapter 3.
Two sides of a triangle have length 3 and 4 centimeters, and the angle between them is 90°. What are the measures of the third side, and of the other two angles?
Solution. Geometry tells us that if we know two sides and an included angle of a triangle, then we ought to be able to find the rest of its measurements.
In this case, we can use the Pythagorean Theorem (see page 7) to tell us that the third side of the triangle has measure 5. But geometry will not tell us the measures of the angles. We will learn how to find them in Chapter2.
Let us now turn our attention to circles.
Example 4 In a certain circle, a central angle of 20° cuts off an arc that is 5 inches long. In the same circle, how long is the arc cut off by a central angle of 40°?
That is, if we double the central angle, we also double the length of the arc it intercepts.
In a certain circle, a central angle of 20° determines a chord that is 7 inches long. In the same circle, how long is the chord determined by a central angle of 40°?
Solution. As with Example 4, we can try to divide the 40° angle into two 20° angles:
However, it is not so easy to relate the length of the chord determined by the 40° angle to the lengths of the chords of the 20° angles. Having doubled the angle, we certainly have not doubled the chord.
In a circle of radius 7, how long is the chord of an arc of 90°?
Solution. If we draw radii to the endpoints of the chord we need, we will have an isosceles right triangle:
Example 7 In a circle of radius 7, how long is the chord of an arc of 38°?
Solution. Geometry does not give us the tools to solve this problem. We can draw a triangle, as we did in Example 6:
But we cannot find the third side of this triangle using only geometry. However, this example does illustrate the close connection between measurements in a triangle and measurements in a circle.
Trigonometry will help us solve all these kinds of problems. However, trigonometry is more than just an extension of geometry. Applications of trigonometry abound in many branches of science.
Look at any pendulum as it swings. If you look closely, you will see that the weight travels very slowly at either end of its path, and picks up speed as it gets towards the middle. It travels fastest during the middle of its journey.
The graph below shows the time of sunrise (corrected for daylight savings) at a certain latitude for Wednesdays in the year 1995.
The data points have been joined by a smooth curve to make a continuous graph over the entire year.
We expect this curve to be essentially the same year after year. However, neither geometry nor algebra can give us a formula for this curve. In
Chapter 8: we will show how trigonometry allows us to describe it mathematically.
Trigonometry allows us to investigate any periodic phenomenon in any physical motion or change that repeats itself.
We will start our study of trigonometry with triangles, and for a while we will consider only right triangles. Once we have understood right triangles, we will know a lot about other triangles as well.
Suppose you wanted to use e-mail to describe a triangle to your friend in another city. You know from geometry that this usually requires three pieces of information (three sides; two sides and the included angle; and so on).
For a right triangle, we need only two pieces of information, since we already know that one angle measures 90°.
In choosing our two pieces of information, we must include at least one side, so there are four cases to discuss:
a) the lengths of the two legs;
b) the lengths of one leg and the hypotenuse;
c) the length of one leg and the measure of one acute angle;
d) the length of the hypotel}use and the measure of one acute angle.
Suppose we want to know the lengths of all the sides of the triangle. For cases (a) and (b) we need only algebra and geometry. For cases (c) and (d), however, algebraic expressions do not (usually) suffice. These cases will introduce us to trigonometry, in Chapter 1.
The Pythagorean theorem
We look first at the chief geometric tool which allows us to solve cases (a) and (b) above. This tool is the famous Pythagorean Theorem.
We can separate the Pythagorean theorem into two statements:
Statement 1: If a and b are the lengths of the legs of a right triangle, and c is the length of its hypotenuse, then
These two statements are converses of each other. They look similar, but a careful reading will show that they say completely different things about triangles.
In the first statement, we know something about an angle of a triangle (that it is a right angle) and can conclude that a certain relationship holds among the sides.
In the second statement, we know something about the sides of the triangle, and conclude something about the angles (that one of them is a right angle).
The Pythagorean theorem will allow us to reconstruct a triangle, given two legs or a leg and the hypotenuse. This is because we can find, using this information, the lengths of all three sides of the triangle.
As we know from geometry, this completely determines the triangle.
In the English university town of Oxford, there are sometimes lawns occupying rectangular lots near the intersection of two roads (see diagram).
In fact, we can make a stronger statement than statement II:
In such cases, professors (as well as small animals) are allowed to cut across the lawn, while students must walk around it. If the dimensions of the lawn are as shown in the diagram, how much further must the students walk than the professors in going from point A to point B?
Solution. Triangle ABC is a right triangle, so statement I of the Pythagorean theorem applies:
Example 11 Show that a triangle with sides 3, 4, and 5 is a right tnangle
Our best friends (among right triangles) There are a few right triangles which have a very pleasant property: their
sides are all integers. We have already met the nicest of all (because its sides are small integers): the triangle with sides 3 units, 4 units and 5 units.
But there are others.
Our next best friends (among right triangles) In the previous section, we explored right triangles with nice sides.
We will now look at some triangles which have nice angles. For example, the two acute angles of the right triangle might be equal.
Then the triangle is isosceles, and its acute angles are each 45o. Or, we could take one acute angle to be double the other. Then the triangle has acute angles of 30 and 60°.
But nobody is perfect. It turns out that the triangles with nice angles never have nice sides. For example, in the case of the 45° right triangle, we have two equal legs, and a hypotenuse that is longer:
Our next best friends (llmong right triungles)
If we suppose the legs are each 1 unit long, then the hypotenuse, measured in the same units, is about 1.414213562373 units long, not a very nice number.
For a 30°right triangle, if the shorter leg is 1, the hypotenuse is a nice length3: it is 2. But the longer leg is not a nice length. It is approximately 1.732 (you can remember this number because its digits form the year in which George Washington was born- and the composer Joseph Haydn).
It also turns out that triangles with nice sides never have nice angles.
If we want an example of some theorem or definition, we will look at how the statement applies to our friendly triangles.
Some standard notation
A triangle has six elements (“parts”): three sides and three angles. We will agree to use capital letters, or small Greek letters, to denote the measures of the angles of the triangle (the same letters with which we denote the vertices of the angles).
To denote the lengths of the sides of the triangle, we will use the small letter corresponding to the name of the angle opposite this side.
Some examples are given below:
Because the angles of any triangle add up to 180°, a triangle can be classified as acute (having three acute angles), right (having one right angle), or obtuse (having one obtuse angle). We know from geometry that the lengths
of the sides of a triangle determine its angles.
How can we tell from these side lengths whether the triangle is acute, right, or obtuse? Statement II of the Pythagorean theorem gives us a partial answer: If the side lengths a, b, c satisfy the relationship
then the triangle is a right triangle. But what if this relationship is not satisfied?
We can tell a bit more if we think of a right triangle that is “hinged” at its right angle, and whose hypotenuse can stretch (as if made of rubber).
The diagrams below show such a triangle. Sides a and b are of fixed length, and the angle between them is “hinged.”
As you can see, if we start with a right triangle, and “close down” the hinge, then the right angle becomes acute. When this happens, the third side (labeled c) gets smaller. In the right triangle, so we can
In the same way, if we open the hinge up, angle C becomes obtuse, and the third side gets longer:
Sowe see that
Proof of the Pythagorean theorem
There are many proofs of this classic theorem. Our proof follows the Greek tradition, in which the squares of lengths are interpreted as areas. We first recall statement I from the text:
If a and b are the lengths of the legs of a right triangle, and c is the length of its hypotenuse, then
Let us start with any right triangle. The lengths of its legs are a and b, and the length of its hypotenuse is c:
We draw a square (outside the triangle), on each side of the triangle:
We must show that the sum of the areas of the smaller squares equals the area of the larger square:
The diagram below gives the essence of the proof. If we cut off two copies of the original triangle from the first figure, and paste them in the correct niches, we get a square with side c:
We fill in some details of the proof below.
We started with an oddly shaped hexagon, created by placing two squares together. To get the shaded triangle, we lay off a line segment equal to b, starting on the lower left-hand comer.
Then we draw a diagonal line.
This will leave us with a copy of the original triangle in the comer of the hexagon:
(Notice that the piece remaining along the bottom side of the hexagon has length a, since the whole bottom side had length a+ b.)
Triangle ABC is congruent to the one we started with, because it has the same two legs, and the same right angle. Therefore hypotenuse A B will have length c.
Next we cut out the copy of our original triangle, and fit it into the niche created in our diagram:
The right angle inside the triangle fits onto the right angle outside the hexagon (at D), and the leg of length a fits onto segment BD, which also has length a.
Connecting A to E, we form another triangle congruent to the original (we have already seen that AF =a, and EF = b because each was a side of one of the original squares).
This new copy of the triangle will fit nicely in the niche created at the top of the diagram:
Why will it fit? The longer leg, of length b, is certainly equal to the upperside of the original hexagon. And the right angles at G must fit together.
But why does G H fit with the other leg of the triangle, which is of length a?
Let us look again at the first copy of our original triangle. If we had placed it alongside the square of side b, it would have looked like this:
But in fact we draw it sitting on top of the smaller square, so it was pushed up vertically by an amount equal to the side of this square, which is a:
So the amount that it protrudes above point G must be equal to a. This is the length of G H, which must then fit with the smaller leg of the second copy of our triangle.
One final piece remains: why is the final figure a square? Certainly, it has four sides, all equal to length c. But why are its angles all right angles?
Let us look, for example, at vertex B. Angle C B D was originally a right angle (it was an angle of the smaller square). We took a piece of it away when we cut off our triangle, and put the same piece back when we pasted the triangle back in a different position. So the new angle, which is one in our new figure, is still a right angle. Similar arguments hold for othervertices in our figure, so it must be a square.
In fact all the pieces of our puzzle fit together, and we have transformed the figure consisting of squares with sides a and b into a square with side c.
Since we have not changed the area of the figure, it must be true that a
Finally, we prove statement II of the text:
If the positive numbers a, b, and c satisfy then a triangle with these side lengths has a right angle opposite the side with length c.
We prove this statement in two parts. First we show that the numbers a, b, and care sides of some triangle, then we show that the triangle we’ve created is a right triangle.
Geometry tells us that three numbers can be the sides of a triangle if and only if the sum of the smallest two of them is greater than the largest.
But can we tell which of our numbers is the largest? We can, if we remember that for positive numbers,
What kind of triangle is it? Let us draw a picture:
Does this triangle contain a right angle? We can test to see if it does by copying parts of it into a new triangle. Let us draw a new triangle with sides a and b, and a right angle between them:
How long is the hypotenuse of this new triangle? If its length is x, then statement I of the Pythagorean theorem (which we have already proved)
to note that this new triangle, which has the same three sides as the original one, is congruent to it. Therefore the sides of length a and b in our original triangle must contain a right angle, which is what we wanted to prove.