Inverse Trigonometric Ratios
Regular or Arc –Students will sometimes be confused about when to use the regular trigonometric function and
when to use the inverse. They understand to concepts, but do not want to go through the entire thought process each
time they must make the decision. I give them this short rule of thumb to help them remember: When looking for a
ratio or side length, use regular and when looking for an angle use arc. They can associate “angle” and “arc” in their
minds. Use the alliteration.
Which Trig Ratio –A common mistake students make when using the inverse trigonometric functions to find angles
in right triangles is to use the wrong function. They may use arcsine instead of arccosine for example. There is a
process that students can use to reduce the number of these kinds of errors.
a. First, the students should mark the angle whose measure is to be found. With the angle in question highlighted,
it is easier for the students to see the relationship the sides have to that angle. It is fun for the students to use
colored pencils, pens, or highlighters.
b. Next, the students should look at the sides with known side measures and determine their relationship to the
angle. They can make notes on the triangle, labeling the hypotenuse, the adjacent leg and the opposite let. If
they are having trouble with this I have them look for the hypotenuse first and always highlight it green, then
they and decide between opposite and adjacent for the remaining to sides.
c. Now, they need to look at the two sides they have chosen, and decide if they need to use sine, cosine, or
tangent. It might help to have a mnemonic device to help them remember the definitions of the trigonometric
functions. A common one issoh-cah-toa. The student can write this abbreviation on the top of every paper
and refer to it when necessary. For example, in an exercise, if they decide it is the adjacent leg to the angle,
and the hypotenuse that they have measures for, that is the“ah”portion ofcah. They will know to use cosine,
and be reminded that the length of the adjacent leg will e in the numerator of the ratio.Make a Graph –Sometimes student will have a hard time seeing a pattern in a list of numbers. One way to help
them remember the general trends in the trigonometric ratios is two have them make a graph. They can put the
angle measure on the horizontal axis and the ratio, in decimal form, on the vertical axis. They will have to use
different scales, of course. Now they can use their calculators to find the trig values of different angles at every five
or ten degrees between zero and ninety and plot points on their graph. The comparison would be most meaningful
if they put all three on the same set of axes with different colors. The process of making the graph and the visual
representation of the pattern will form an impression in the students’ minds that will be useful and lasting.
Acute and Obtuse Triangles
Law of Sines or Law of Cosines –At first, it may be difficult for student to determine if they need to use the Law
of Sines or the Law of Cosines to find a measure in a particular situation. Here is a good thought process for them
to use.
a. First have them look for the two, fairly easy to recognize, Law of Cosines situations. They have all three sides
and are looking for an angle, or have two angles and the included side and are looking for the third side.
b. If it is not one of these, then they need to try to set-up a Law of Sines proportion.The Third Angle –Remind students that the three angles of a triangle have a sum of 180 degrees, and that this fact
is often helpful when applying the Law of Sines or Cosines. Sometimes they may not be able to fine the angle they
want directly, but if they find the third angle, they can use the Triangle Sum Theorem to get the measure they need.
Two Exercises in One -Sometimes the students will have to use both the Law of Sines and the Law of Cosines to
find a measure in a specific triangle. For instance, let’s say they have two sides and the included angle of a triangle,
2.8. Right Triangle Trigonometry