TABLE2.13:(continued)
45 − 45 −90 Triangle 30 − 60 −90 Triangle
b=x√
3
Derive with Variables –The beginning of the last chapter offers students a good amount of experience with ratios.
If they did well on those sections, it would benefit them to see the derivation of the ratios done with variable
expressions. It would give them practice with a rigorous derivation, review and apply the algebra they have learned,
and help them see how the triangles can change in size.
Exact vs. Decimal Approximation –Many students do not realize that when they enter
√
2 into a calculator and
get 1.414213562, that this decimal is only an approximation of
√
- They also do not realize that when arithmetic
is done with an approximation, that the error usually grown. If 3.2 is rounded to 3, the error is only 0.2, but if the
three is now multiplied by five, the result is 15, instead of the 16 it would have been if original the original number
had not been rounded. The error has grown to 1.0. Most students find it more difficult to do operations with radical
expressions than to put the numbers into their calculator. Making them aware of error magnification will motivate
them to learn how to do operations with radicals. In the last step, it may be nice to have a decimal approximation so
that the number can be easily compared with other numbers. It is always good to have an exact form for the answer
so that the person using your work can round the number to the desired degree of accuracy. Less accuracy is needed
for building a deck than sending a robot to Mars?
Tangent Ratio
Trig Thinking –Students sometimes have a difficult time understanding trigonometry when they are first introduced
to this new branch of mathematics. It is quite a different way of thinking when compared to algebra or even geometry.
Let them know that as they begin their study of trigonometry in the next few sections the calculations won’t be
difficult, the challenge will be to understand what is being asked. Sometimes students have trouble because they
think it must be more difficult than it appears to be. Most students find they like trigonometry once they get the feel
of it.
Ratios for a Right Angle –Students will sometimes try to take the sine, cosine or tangent of the right angle in a
right triangle. They should soon see that something is amiss since the opposite leg is the hypotenuse. Let them
know that there are other methods of finding the tangent of angles 90 degrees or more. The triangle based definitions
of the trigonometric functions that the students are learning in this chapter only apply to angles in the interval
0 degrees<m<90 degrees.
The Ratios of anAngle –The sine, cosine, and tangent are ratios that are associated with a specific angle. Emphasize
that there is a pairing between an acute angle measure, and a ratio of side lengths. Sine, cosine, and tangent
is best described as functions. If the students’ grasp of functions is such that introducing the concept will only
confuse matters, the one-to-one correspondence between acute angle and ratio can be taught without getting into
the full function definition. When students understand this, they will have an easier time using the notation and
understanding that the sine, cosine, and tangent for a specific angle are the same, no matter what right triangle it is
being used because all right triangles with that angle will be similar.
Use Similar Triangles –Many students have trouble understanding that the sine, cosine, and tangent of a specific
angle measure do not depend on the size of the right triangle used to take the ratio. Take some time to go back and
explain why this is true using what the students know about similar triangle. It will be a great review and application.
Sketchpad Activity:
a. Students can construct similar right triangles using dilation from the transformation menu.2.8. Right Triangle Trigonometry