The sine and its relatives we have studied four different trigonometric ratios: sine, cosine, tangent, and cotangent. These four are closely related, and it will be helpful to explore their relationships. We have already seen thatfor any acute angle a. The following examples introduce us to a number of other relationships.
Example 17 If sin a = 3/5, find the numerical value of cos a, tan a, and cot a.
Solution. The fraction 3/5 reminds us of our best friend, the 3-4-5 triangle:
In fact, a is the measure of one of the angles in such a triangle: the one opposite the side of length 3 (see the diagram above). Having drawn this triangle, we easily see that cos a= 4/5, tan a= 3/4, and cot a= 4/3.
Example 18 If sin a = 2/5, find the numerical value of cos a, tan a, and cot a.
Solution. We can again draw a right triangle with angle a:
Example 19 We can find an answer to the question in Example 18 in a different way. For the same angle a, we have the right to draw a different right triangle, with hypotenuse 1, and leg 2/5. Do the calculation in this case for yourself. It will produce the same result.
Example 20 If sin a = a, where 0 < a < 1, express in terms of a the value of cos a, tan a, and cot a.
Solution. As before, we choose a right triangle with an acute angle equal to a:
l’he sine und its relutives.
we know all the sides of this triangle, and we can write everything down easily:
Algebra or geometry
Example 21 Suppose sin a = 1/2. Find the numerical value of cos a, tan a, and cota.
Solution. We can do this geometrically, by drawing a triangle Or we can do this algebraically, using the results of
Example 20. For instance,
Or did you notice right away that a is an angle in one of our friendly triangles?
You may have some objection to taking the hypotenuse of our triangle to have length
1. If you insist, we can take some length c for this hypotenuse. We will then get the same results, but the calculations will be longer. For example, suppose sin a = x. Choose a right triangle that contains a, and that has hypotenuse of length c. Suppose the leg opposite a has length a. Then a I c = x, since sin a = x. So a = ex. If we are asked for cos a, we can suppose the length of the other leg is b .
Another way to prove a new identity is to show that it follows from other identities that we know already.
Example 23 Prove the identity tan a cot a = 1.
Solution. From our table, we see that tan a = sin a I cos a. We also see that cot a = cos a I sin a. Therefore,
You will have a chance to practice both these techniques in the exercises below.
Identities with secant and cosecant While we do not often have to use the secant and cosecant, it is often convenient to express the fundamental identities above in terms of these two ratios. We can always restate the results as desired, using the fact that sec a = 1/ cos a and esc a = 1/ sin a.
This last identity was proven in Example 24, page 47.
You may remember from geometry that the hypotenuse is the largest side in a right triangle (since it is opposite the largest angle). So the ratio of any leg to the hypotenuse of a right triangle is less than 1. It follows that sin a < 1 and cos a < 1 for any acute angle a.
That is all the background you need to do the following exercises.
Calculators and tables
It is, in general, very difficult to get the numerical value of the sine of an angle given its degree measure. For example, how can we calculate sin 19°?
One way would be to draw a right triangle with a 19° angle, and measure its sides very accurately. Then the ratio of the side opposite the angle to the hypotenuse will be the sine of 19°.
But this is not a method that mathematicians like. For one thing, it depends on the accuracy of our diagram, and of our rulers. We would like to find a way to calculate sin 19° using only arithmetic operations. Over the centuries, mathematicians have devised some very clever ways to calculate sines, cosines, and tangents of any angle without drawing triangles.
We can benefit from their labors by using a calculator. Your scientific calculator probably has a button labeled “sin,” another labeled “cos,” and a third labeled “tan.” These give approximate values of the sine, cosine, and tangent (respectively) of various angles.
Warning: Most “nice” angles do not have nice values for sine, cosine, or tangent. The values of tan 61 o or sin 47° will not be rational, and will not even be a square root or cube root of a rational number. There are a very few angles with integer degree measures and “nice” values for sine, cosine, or tangent.
Getting the degree measure of an angle from its sine
Example 26 What is the degree measure of the smaller acute angle of a right triangle with sides 3, 4, and 5?
Solution. We could draw a very accurate diagram, and use a very accurate protractor to answer this question. But again, mathematicians have developed methods that do not depend on the accuracy of our instruments.
Your calculator uses these methods, but you must know how the buttons work.
Solving right triangles
Many situations in life call for the solution of problems like the following.
Example 27 The hypotenuse of a right triangle is 5, and one of its acute angles is 37 degrees. Find the other two sides.
Because the sun is so far away from the earth, the rays of light that reach it from the earth are almost parallel. If we think of a small area of the earth as fiat (and we usually do!), then the sun’s rays strike this small region at the same angle:
So we can, for example, tell how long the shadow of an object will be, given its length.
Example 28 If the rays of the sun make a 23 o angle with the ground, how long will the shadow be of a tree which is 20 feet high?
Solution. In the diagram below, AC is the tree, and BC is the shadow:
Another approach to the sine ratio
There is a simple connection between the sine of an angle and chords in a
Theorem If a is the angle subtended by a chord P B at a point on a circle of radius r (such as point A in the diagram below), then
Before we prove this theorem, let us resolve a problem in the way it is stated. We can pick different points on the circle (such as A’ and A” in the figure below), and consider the different angles subtended by the same chord PB:
Does it matter which point we pick? No, it does not. An important theorem of geometry asserts that all the angles subtended by a given chord in a P B we choose.
For this reason, we can prove our theorem by making a very special choice: for point A, we choose the point diametrically opposite to point B:
The same geometric theorem about inscribed angles assures us that LAP B is a right angle, so sin ex
One natural way to measure an arc of a circle is to ask what portion of its circle the arc covers. We can look at the arc from the point of view of the center of the circle, and draw the central angle that cuts off the arc:
Inscribed angles and their arcs
An important theorem of geometry relates the degree-measure of an arc not to its central angle, but to any inscribed angle which intercepts that arc:
Theorem The degree measure of an inscribed angle is half the degree measure
of its intercept arc.
Proof We divide the proof into three cases.
First we prove the statement for the case in which one side of the inscribed angle is a diameter.
Suppose the center of the circle is not on one side of the inscribed angle, but inside it.
Suppose the center of the circle is outside the inscribed angle.
As a corollary to the theorem above, we state Thales’s theorem, one of the oldest mathematical results on record:
Theorem An angle inscribed in a semicircle is a right angle.
The proof is a simple application of the previous result, and is left for the reader as an exercise.
If we have a particular object, which we will represent as a line segment, we are sometimes not so much interested in how big it is, but how big it looks. We can measure this by seeing how much of our field of vision the object takes up. If we think of standing in one place and looking all
For example, viewed from the earth, the angle subtended by a star is very, very small, although we know that the star is actually very large. And the angle subtended by the sun is much greater, although we know that the sun, itself a star, is not the largest one.
Suppose the angle subtended by object AB at P measures 60°. Can we find other points at which AB subtends the same angle? From what positions does it subtend a larger angle? From what positions a smaller angle?
The answer is interesting and important. If we draw a circle through points A, Band P, then AB will subtend a 60° angle at any point on the circle, to one side of line AB:
Also, A B will subtend an angle greater than 60° at any point inside the circle (to one side of line AB), and will subtend an angle less than 60° at any point outside the circle.
All this follows from the converse theorem to the one in the previous section:
Theorem Let A B subtend a given angle at some point P. Choose another point Q on the same side of the line AB as point P. Then
• If A B subtends the same angle at point Q as at point P, then Q is
on the circle through A, B and P.
• If AB subtends a greater angle at Q, then Q is inside the circle
through A, P and B.
• If AB subtends a smaller angle at Q, then Q is outside the circle
through A, P and B.
(Remember that Q and P must be on the same side of line A B.) The proof of this converse will emerge from the exercises below.