Reciprocal, Quotient, Pythagorean
This section reviews the definitions of the trig functions and the Trigonometric Pythagorean Theorem. The Pythagorean identities, you’ll recall, were first covered in lesson 1.6, but here we see a slightly different way of deriving them. It may be useful to reinforce knowledge of the identities by walking through the derivations, but for many students this will simply be review.
Confirm Using Analytic Arguments
The diagram here with the vertical line representing a distance of t units may confuse students a little, but it simply demonstrates in a slightly unusual way the fact that any real number can correspond to a distance traveled around the unit circle, and therefore to an angle on that circle. Again, this should be review for most students.
Confirm Using Technological Tools
We see here that knowing just one trig function of an angle does not uniquely determine the angle, as there are always two quadrants it could be in (which two depends on whether the value of the trig function is positive or negative). Generally, knowing a second trig function of the angle, or at least knowing its sign, will narrow down which quadrant the angle could be in—but note that this won’t be the case if the second function is just the reciprocal of the first function. So, for example, knowing the signs of the tangent and cotangent functions won’t tell us what quadrant the angle is in, because the tangent and cotangent always have the same sign whatever the quadrant—and the same is true for the sine and cosecant, or the cosine and secant. But knowing the signs of, say, the tangent and secant functions will tell us which quadrant the angle is in, and the same is true for any two trig functions that are not each other’s reciprocals.
Working with Trigonometric Identities
Students may get a little confused by the fact that the equations we’re working with reduce to seemingly obvious identities. The point here, of course, is that we start with an equation that declares two complex expressions to be equivalent, and then prove the equation is true by showing it can be changed into a form that is much more obviously true.
And that’s exactly what we’re doing in proving these trig identities: by showing that both sides of the equation reduce to the same expression, we’re proving that the equation is always true for any value of θ. And once we know it’s true, we can use it to make useful substitutions when solving trig problems in the future.
Calculators can be useful in verifying identities, but it is dangerous to rely on them too much. If the graphs of two expressions look identical, it may mean the expressions are indeed equivalent, but it may also mean that the difference between them is just too small for the graph to show, or that they are only equivalent over this small interval. Since a graphing calculator can only show us part of a graph and can only draw it with limited precision, it cannot tell us for sure if two expressions really are mathematically equivalent.
What a graph can do, though, is tell us for sure if two expressions are not equivalent. If the graphs of the expressions look wildly different—in fact, if they look even a little bit different—then we can safely say that the expressions are not equal.
Sum and Difference Identities for Cosine
Difference and Sum Formulas for Cosine
Use Cosine of Sum or Difference Identities to Verify Other Identities
Call attention to the labels “Identity A” and “Identity B” here. These labels aren’t official names for these identities, but they will be used to refer back to them later in this lesson.
The identities themselves simply say that the sine of an angle is equal to the cosine of the angle’s complement, and vice versa. We’ve already seen that this is true from working with angles in right triangles, as the sine of one acute angle in a triangle is equal to the cosine of the other, and the two angles are each other’s complements.
Use Cosine of Sum or Difference Identities to Find Exact Values
As before, note that calculators cannot tell us with absolute certainty if two expressions have identical values; they might, for example, be identical up to twenty decimal places, but differ after that point. However, if two expressions seem to have equal values when plugged into a calculator, there is a reasonably good chance that they are really equal—and more importantly, the calculator will clearly tell us if they are not equal at all. Hence, double-checking answers with a calculator is still a good idea.
Sum and Difference Identities for Sine and Tangent
Sum and Difference Identities for Sine
(For reference, Identities A and B are on page 244.)
Sum and Difference Identities for Tangent
Once again, expressing the sum formula for tangent in words may make it easier for students to remember: “The tangent of the sum equals the sum of the tangents over 1 minus the product of the tangents.” There isn’t really a concise way to do this for the sine formulas, though.
Consider whether you want your students to memorize these and other trig identities or not. They definitely need to develop a good sense for when to use which ones, but the formulas themselves are the least important part of that knowledge; in fact, knowing how to derive the formulas may be more useful than simply knowing the formulas themselves. Instead of requiring that all the formulas be memorized for a test, for example, it might be more educational to supply a few of the formulas and require that students re-derive the others from the few given (after making sure, of course, that the formulas supplied are sufficient to derive the others from.)
Of course, some students might find that a little too challenging, but on the other hand, some will find it easier than memorization. Perhaps a good compromise would be to provide a few of the formulas so that students can derive the others if they want to, but not make that derivation a required part of the exam. Since students will presumably need to use any or all of the formulas to solve some of the exam questions, they will still have to either derive the formulas or have them memorized, but can do whichever of those two works best for them.
Deriving the Double-Angle Identities
Applying the Double-Angle Identities
Finding Angle Values Given Double Angles
Simplify Expressions Using Double-Angle Identities
In the example given here, you can show, if you wish, how choosing a different double-angle formula would make the expression we are working with more complicated, and hence why the one used here is the best one to use in this case.
The first paragraph here describes a useful strategy for getting an idea of what sort of answers are reasonable before attempting to solve a trig problem: figure out, if you can, approximately where the given angles are and approximately what the values of the trig functions should be for those angles, or at least figure out upper or lower bounds on the trig functions based on whether the given angle is greater or less than an angle you are already familiar with, and whether the relevant trig functions are increasing or decreasing in the part of the unit circle where the angle is located.
Deriving the Half-Angle Formulas
Use Half-Angle Identities to Find Exact Values
Find Half-Angle Values Given Angles
Using the Half- or Double-Angle Formulas to Verify Identities
Product-and-Sum, Sum-and-Product and Linear Combinations of Identities
Transformations of Sums, Differences of Sines and Cosines, and Products of Sines and Cosines
For an extra challenge, you might ask students to derive the three formulas whose derivations are not shown, by applying similar reasoning to that used in the derivation that is shown.
Transformations of Products of Sines and Cosines into Sums and Differences of Sines and Cosines
The key thing they need to understand is that each of these identities is a tool to be used in different situations, depending on what knowledge they already have. If they need to find the sine or cosine of an angle, and that angle can be expressed as a sum or difference of two angles of which they already know the sine and cosine, then the sum formula from earlier is useful. If they need to know the product of two sines or cosines, and they don’t know the sines or cosines themselves, but do know the sine and cosine of the sum and difference of those two angles, then the product formula learned here is useful. In general, they should form the habit of writing down exactly what it is they are looking for and then considering which tools in their possession might apply to that particular situation.