b. After choosing a specific angle they should measure the corresponding angle in all the triangles. Each of these
measurements should be equal.
c. The legs of all the right triangles can be measured.
d. Then the tangents can be calculated.
e. Student should observe that all of ratios are the same.Remind the students that if the right triangles have one set of congruent acute angles, then they are similar by the AA
Triangle Similarly Postulate. Once the triangles are known to be similar it follows that their sides are proportional.
The ratios are written using two sides of one triangle and compared to the ratios of the corresponding sides in the
other triangle. This is different but equivalent to the ratios students probably used to find missing sides of similar
triangles in previous sections.
Sine and Cosine Ratio
Trig Errors are Hard to Catch –The math of trigonometry is, at the point, not difficult. Not much computation
is necessary to chose two number and put them in a ratios. What students need to be aware of is how easy it is to
make a little mistake and not realize that there is an error. When solving an equation the answer can be substituted
back into the original equation to be checked. The sine and cosine for acute angles do not have a wide range. It is
extremely easy to mistakenly use the sine instead of the cosine in an application and. The difference often is small
enough to seem reasonable, but still definitely wrong. Ask the student to focus on accuracy as they work with these
new concepts. Remind them to be slow and careful.
Something to Consider –Ask the students to combine their knowledge of side-angle relationships in a triangle with
the definition of sine. How does the length of the hypotenuse compare to the lengths of the legs of a right triangle?
What does that mean about the types of numbers that can be sine ratios? With leading questions like these students
should be able to see that the sine ratio for an acute angle will always be less than one. This type of analysis will
prepare them for future math classes and increase their analytical thinking skills. It will also be a good review of
previous material and help them check there work when they first start writing sine and cosine ratios.
Rationalizing the Denominator –Sometimes student will not recognize that √^1
2
and√
2
2 are equivalent. Most
likely, they learned how to rationalize denominators in algebra, but it is nice to do a short review before using these
types of ratios in trigonometry. Student will have to be able to easily switch between the two forms of the number
when working with the unit circle in later classes.
Two-Step Problems –Having the students write sine, cosine, and tangent ratios as part of two-step problems will
help them connect the new material that they have learned to other geometry they know. They will remember it
longer, and be better able to see where it can be applied.
Key Exercise:
- 4 ABCis a right triangle with the right angle at vertexC.
AC=3 cm andBC=4 cm
What is the sign of^6 A?
Answer:AB=5 cm by the Pythagorean theorem, therefore sinA=^45.
Note: The sine of an angle does not have units. The units will cancel out in the ratio.
Chapter 2. Geometry TE - Common Errors