Since right angles are so common in our culture - and in any case any non-right-angled triangle can be broken up into right angled pieces, it is particularly useful to know the ratios of sides of right angled triangles. These are the famous trigonometric ratios - which we will now consider.
When you have finished this section you should be able to:
· State the definitions of the six trig ratios
· Find the values of all six trig ratios for an any angle in a right angled triangle whose side lengths are specified (or even if just two are)
· Recognize special cases of angles whose trig ratios can be determined exactly in terms of radicals by using basic geometry
· Use a calculator or tables to determine approximate trig ratios for any given angle
· Determine all of the sides of a right triangle given one side and one of the acute angles
1. Read section 5.2 of the text.
3. Read the following Study Notes and Discussion.
4. Follow the instructions regarding Practice Exercises.
Definitions
Although there are six possible trigonometric ratios, they are all related.
For example, , and the “reciprocal identities” in the box at top of p363, show that all the others can also be expressed in terms of sin and cos.
Perhaps the most efficient way to remember all of these definitions and relationships is to start by fixing in your mind the fact that for an acute angle the sine and cosine both involve dividing one of the two perpendicular sides by the hypotenuse, and that its the sine that uses the opposite side (and that the cosine uses the one that connects with the angle).
For non-acute angles the extended definition given on page 371 relates the cosine to a ratio involving the x-coordinate of a point on the terminal side of the angle. If the point is at distance from the origin, then the cosine is the x-coordinate, and another way to remember this might be as “cos goes across”.
Then memorize tan=sin/cos. It should be obvious that the hypotenuse cancels to give opp/adj, and that geometrically this corresponds to the slope of the hypotenuse if the angle is in standard position.
Finally, think of the other three to be defined as the reciprocals.
Note: It is important to get in the habit of calling these reciprocals rather than inverses. They are not “inverse functions” of the kind we discussed in Unit 2 (and in Unit 4 relating exponentials and logarithms). So be sure to avoid the -1 exponent notation. Remember, this notation is reserved for the composition inverse, or “inverse function”, NOT the reciprocal.
Special Cases
The special cases identified in Example 2 of the text are important.
The best way to become familiar with them is to recall the geometric argument every time you use one of them until it becomes “burned in”.
Note: One thing to avoid is lazily omitting the reference to degrees when remembering for example that .This will definitely lead you astray, since in the next Unit (and in Calculus ) you will be expected to use the convention that when the units are not specified they are assumed to be radians.
For angles other than these special cases people developed clever ways of getting approximate values to a high degree of accuracy. Historically they recorded the results in tables and nowadays similar methods are implemented on your calculator when you use one of the trig buttons.
Applications
The effort of computing trig tables was justified by the fact that quick angular measurements could be used by surveyors and navigators in place of often less convenient distance measurements.
e.g. A surveyor whose eye level is 1.5m above the ground observes the top of a tree at a distance of 100m to be at an angle of above the horizon. How tall is the tree?
(see answer #1)
By using trigonometry, the forest company has managed to use just one distance or length rather than the three that were used for a similar problem back in the first section of this unit, and now has just one person do the work that previously required two!
Basic Identities
Although there are six possible trigonometric ratios, we have observed that they are all related. In particular, using tan=sin/cos and the “reciprocal identities” we saw that all the others can be expressed in terms of the sine and cosine.
Furthermore, these two are related by the Pythagorean Identity:
.
So we can express cos in terms of sin and vice versa:
(For the acute angles that we have been considering so far, only the +sign applies.)
So there is really essentially only one independent ratio to be found for each angle.
Also, since the angles of a triangle must always add to 180° we see that if one of the non right angles is fixed at a° then the other is also fixed - at (90-a)°, which is called the 'complement' of a°. Each trig ratio for a° is therefore a (different) trig ratio for (90-a)°. In fact the cosine of an angle is just the sine of its complement (and vice-versa, etc.). This allows us to further reduce the number of independent values, but before computers the construction of a complete set of tables was still a big job.
These identities will be useful to us for simplifying and comparing expressions.
For example if one way of solving a problem gives a result of the form , and another gives , then although at first sight they look different, they are actually both saying the same thing. Can you see why? (see answer #2)
Check your understanding and practice for speed by working through some of the Exercises on pp385-389.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort)
As a minimum you should do #7,13,19,21,23,29,33,39,49,59.