By now, many of your students may be all excited about trigonometry. You’ve managed to persuade them that it is worth to put the efforts to comprehend the technicalities that you will explain to them. But you will also need to remind them of the importance of prerequisite knowledge. To effectively communicate the technicalities to your students, you will need to make some assumptions about their knowledge of some facts. If you choose not to do that, you may run into complications to explain basic results, such as $$1/\sqrt{3}$$ is equal to $$\sqrt{3}/3$$. That they know some algebra and geometry probably is a safe assumption. If your students use a textbook, facts about algebra and geometry may be included. Otherwise, the web may be a good place to start.

A good topic to start with may be angles. However basic angles may be, some of your students may not know how well they will need to know about angles to perform basic trigonometric calculations. So bring the angles in. Explain to them in an intuitive way what angles are; show them many kinds of angles: acute, obtuse, right. Draw many of them. Even at this stage, you may need to start encouraging them to practice with you. Ask them to draw couple angles. Next, bring in measurement of angles. Again measurements will be prominent in future topics. You may need to remind them about units of measurement; in the case of angles, there are several of them, such as radians, degrees, minutes, and seconds. Already at this stage, some of them may be confused. You may need to devise some activities with the units and encourage them to participate, so that they could have a feel for measuring angles.

# A note on practice

Already at this stage, you may need to remind your students that mastery comes with practice. Many of them probably know about that, but it wouldn’t hurt to reiterate this advice. If some of them already have lacunae at this basic stage, they may find the future topics difficult. Many of your students may be able to find the appropriate materials to practice from; if not, you would need to provide some materials with practice problems. If you use a textbook, there are many practice problems. If you teach at a school, you may tell them to turn in the solutions to some of the practice problems for a grade.

# Augmented Reality

Remember we’ve brushed on the importance of technology in mathematics education. There are many technological tools, many of which are open source, that you can use to show vivid illustrations of shapes and angles. At this early stage, it may be in your students’ interest to “see” what the trigonometric notions are before delving into more abstract notions. Augmented Reality (AR) tools have been incredible at doing that. GeoGebra AR is one such tool. There probably are tons of YouTube videos that can give you ideas about which tool(s) to use; see this one, for example.

# Motivating the right triangle

Before introducing the trigonometric functions, you may need to spend some time mentioning the right triangle. It is true that many of your students may already know about the right triangle from their knowledge of geometry, but it wouldn’t hurt to do so. You probably can show them what a right triangle is, by drawing one or by showing concrete examples, such as a square sheet of paper folded diagonally. Then you can give them a formal definition.

For many of your students, the sine of an angle is a new concept, so you need to spend some time on the concept before doing computational examples. Remember that trigonometric functions will be ever present in future topics, so your students need to be accustomed to them as much as possible. There probably are many ways you can introduce the sine of an angle; the makeup of your students may influence the approach you choose. One approach is to introduce the sine of an angle as just a ratio of the length of the side opposite to this angle over the length of the hypothenuse. An advantage of this approach is that it depends on an algebraic notion of ratio that many of your students may already know from their knowledge of algebra. A disadvantage is that it does not give a sense of what the sine of an angle measures. If you have students with a practical bent, then they may be uninspired by this algebraic approach. What about this one? The sine of an angle measures the steepness of the hypothenuse against the base of the right triangle, so the sine of an angle increases as that angle increases from 0 degree to 90 degrees, or from 0 radian to $$\pi/2$$ radian. If you use this approach, remember to draw triangles and show them what happens when the opposite side changes. There’s a nice interactive module on GeoGebra that illustrates the sine of an angle.

You probably need to take as much time to introduce the tangent and secant of an angle. Again, use concrete examples as you explain them. It would never hurt to draw or show many right triangles.

# Cosine, cotangent, and cosecant

By now, you’ve probably realized we haven’t mentioned the cosine, cotangent, and cosecant of an angle. You can choose to introduce them like the sine, tangent, and secant. But you probably can take another route if you introduce or remind your students about complementarity. The two acute angles of a right triangle complete a right angle or they add up to 90 degrees or $$\pi/2$$ radian. This seemingly innocent property comes with some nice results. Let us show you an example. Let us label the two acute angles of a right triangle as $$\alpha$$ and $$\beta$$. Then the cosine of $$\alpha$$ can be defined as the sine of $$\beta$$. Of course, that’s equivalent to defining the cosine of $$\alpha$$ as the ratio of the length of the side adjacent to $$\alpha$$ over the length of the hypothenuse. One advantage of the first definition is that it exhibits the relationship between the sine and the cosine, whose names already suggest such relationship. Another one is that it takes less work to define the cosine since much work has already been done for the sine.

You can use a similar approach for the cotangent and cosecant, and then you may add that these “complementary” functions also are ratios of lengths.

# It’s time for practice

At this point, you probably need not mention any more details about these trigonometric functions. It may be best starting working on some problems. First, show your students how to work out some problems; then encourage them to work on other ones with you. Along the way, some additional details may be needed, but you can explain them on a case-by-case basis. For instance, you can briefly explain the Pythagorean Identity, if needed; it’s a consequence of the Pythagorean Theorem and the definition of sine and cosine as ratios.

To help your students have a strong grounding in the basics, you may need to encourage them to work on more problems on their own, starting with easy ones then to harder ones. If you teach at a school, you may encourage them to read the text, if one is adopted, and assign some problems as homework and collect some for a grade.

# On proof

Even at this basic level, you may need to use certain identities, such as the Pythagorean Identity, and you may need to justify to your students why such identities hold; at least the curious ones might want to know. Providing an explanation to them may help them better understand trigonometry. If an explanation requires only knowledge of geometry, algebra, and knowledge of previous trigonometric facts, then you probably can tell them why an identity holds. For instance, the Pythagorean Identity merely follows from the Pythagorean Theorem and the definitions of sine and cosine as ratios. Also, if you have students interested in studying more advanced mathematics, then justifying identities cannot hurt them.