After you’ve spent some time practicing with your students and give them some time to work on exercises on their own, you may need to move on to other topics.

Now that you’re done working with evaluating trigonometric functions on an acute angle, it’s probably time to introduce evaluating trigonometric functions on an arbitrary angle. Much of the trigonometry you’ll cover may involve non-acute angles, so it’s wise to bring them in.

Non-acute angles

You may need to emphasize to your students the sine of an angle can be evaluated not only for acute angles but for any other angles: angles as great as 1 million degrees and as small as 1 millionth of a degree. First, you may need to show them what those arbitrary angles look like. You may need to show them that a convenient way to represent angles is to draw a ray starting from the origin of a rectangular coordinate system, and any angle can be represented by the angle between that ray and the axis of positive abscissas. There are two nice related demos on GeoGebra; you can find them here and here. You may also explain that values of angles are positive when measurement starts from the axis of abscissas to the ray in a counterclockwise way and values are negative when measurement is performed in a clockwise way. Measuring large angles may be the new trick, so show them how it’s done: a large number of revolutions correspond to a large angle. For instance, about 2,777.78 revolutions correspond to an angle of 1 million degrees. Furthermore, you may show them that small angles are obtained by rotating the ray very close to the axis of positive abscissas.

Generalizing the sine function

To define the sine of any angle, we think it’s best to start with a ray of arbitrary length instead of a ray of unit length. The first alternative may show your students how the sine is defined is a much more general way, and the unit circle can be understood as just a special case, with some nice properties. After all, when defining the sine of an acute angle of a right triangle, there is no requirement that the hypothenuse be of unit length, so we don’t see why we should impose that requirement right at the beginning for the more general definition of the sine.

First, explain what a reference angle is. Then you may show them that given any reference angle made by a ray of arbitrary length, a right triangle may be drawn with that ray as the hypothenuse and the height as a segment starting from the tip of the ray and perpendicular to the axis of abscissas; you can see an example here. You may briefly mention that the tip of the ray can be located with a coordinate made of two real numbers, the abscissa and the ordinate, and these real numbers can be negative, zero, or positive. Then you may define the sine of the reference angle as the ratio of the ordinate over the length of the ray. Defining the sine as such may be familiar to them since you’ve already done so in the case of the right triangle. The slight novelty might be that the sine of the reference angle can be negative or zero since the ordinate of the tip of the ray can be negative or zero.

Furthermore, you may mention to them that any angle made by a ray of arbitrary length has an associated reference angle. You may need to justify this claim to them; a geometric justification is straightforward, which is that, for any angle, the ray either falls in a quadrant or coincides with one of the axes. If you have curious students, they may want to know why this is the case. You may also need to show them some examples: the reference angle associated to an angle of 30 degrees is an angle of 30 degrees; the reference angle associated to an angle of 320 degrees is an angle of 40 degrees; and the reference angle associated to an angle of 3000 degrees is an angle of 60 degrees. For large angles, explain to them how you find the associated reference angle. Maybe encourage them to organize themselves in small groups, and give them couple angles to find their associated reference angles.

Then you can explain to them that the sine of any angle is the sine of its associated reference angle. During all the working leading to this explanation, you may need to proceed as slowly as possible and encourage them to ask questions along the way. Concepts are important, so it wouldn’t hurt spending some time on explaining them.

Tangent, secant, and so on

When you’re done with the general definition of the sine, you can briefly show them how to define the tangent and secant of any angle. Most of the work has been done with the sine, so you can move on quickly, but still encourage them to ask questions. When it comes to cosine, cotangent, and cosecant, you may proceed as we’ve outlined in here or define them like the sine.

After all these concepts, you may show them how to make some calculations and then encourage them to work on couple calculations in small groups or on their own.

Now the unit circle

Now that all this work has been done for rays of arbitrary lengths, you may mention some of the advantages of using a ray of unit length to define the sine of any angle. For instance, when the ray is of unit length, the sine of the angle made by that ray and the axis of abscissas is just the ordinate of the tip of the ray; remember we’re working in a rectangular coordinate system, so the tip of the ray has a coordinate made of an abscissa and an ordinate. Also, the cosine of that angle is just the abscissa. So, the unit circle simplifies computations.

Moreover, working with the unit circle, or more generally with the rectangular coordinate system, will be helpful when it will be crucial to know if the value of a trigonometric function is negative, zero, or positive. So, encourage your students to work on many exercises involving the unit circle, or more generally the rectangular coordinate system.

The sine as a function on the real numbers

Now that you’re done with explaining what the sine of any angle is, you may tell your students that the sine is a function that takes any real number as a variable, and since the value of a sine of an angle is a ratio of real numbers, then you can explain to them that the sine is a function from the real numbers to the real numbers. Then you may add that the tangent, secant, cosine, cotangent, cosecant are all functions.