Integrating trigonometric functions inquiry

The prompt

Mathematical inquiry processes: Reason; identify and generalise patterns; analyse mathematical structure. Conceptual field of inquiry: Integration by substitution and by parts; trigonometric identities; transformation of graphs.

The inquiry developed out of a discussion in a year 13 lesson. A question in a textbook exercise required students to integrate the trigonometric function in the prompt. The class proposed different ways to solve the problem.

As an inquiry prompt the integrand and integral are intriguing. Knowing that the derivative of -cos(x) is sin(x), students do not expect to see the integral of a sine function in the form of another sine function.

Equivalence

The inquiry also draws upon students' prior learning about trigonometric identities to show that the integrand is equivalent to cos(2x). The first approach below uses identities derived from a translation of the sine graph and a reflection of the cosine graph; the second approach uses an angle difference identity.

Deriving the graph of cos(2x) by transforming the graph of sin(90o - 2x) is often more challenging. The transformations have to be carried out in a specific order, which can be summarised as 'outside in'. However, students often work in the opposite direction - that is,

Sin 2x Stretch by a factor 1/2 parallel to the x-axis;
Sin (-2x) Reflection about the y-axis; and
Sin(90o - 2x) Translation 90o left.

They find that the graph is 45o to the right of where it should be. That is because the translation in step 3 replaces the x in the function with (x + 90o). As a result, the graph of sin(x) is transformed to that of sin(-2(x + 90o)) or sin(-2x - 180o).

The correct order starts with the translation. Then follows the reflection and stretch (see the illustration below) - although, as the graph is symmetrical about the y-axis after the translation, the reflection and stretch can be carried out in either order.

An alternative approach is to rewrite the function in the prompt as sin(-2(x - 45o)), in which case the translation is the final transformation.

Alternative prompt

In the question, notice, and wonder phase of the inquiry, students might assert, show, or prove the equivalence of the integrand to cos(2x). If that is the case, the class will verify the truth of the equation quickly. The prompt is unlikely to stimulate students' curiosity and the inquiry will be in danger of coming to a premature end.

The class would find this prompt more intriguing;

Students' questions include:

Are the integrals correct? How do we verify them?
Is there a (number) pattern in the way the integrals are made up?
How many cases would we need to identify a pattern or be sure a pattern did not exist?
Could we work out an integral for the nth case?

After the students have verified each integral for themselves (see the mathematical notes), they search for a pattern. If they use an online calculator to generate the integrals, they should still explain how the functions with higher exponents are integrated. This might be done through class presentations in which students justify their chains of reasoning.

This line of inquiry comes an end when students realise that, as there is no general method of writing the sine function in terms of cos(x) for different exponents, it is not possible either to find an integral for the nth case or, it follows, to identify a consistent pattern in the integrals.

January 2026

Lines of inquiry

Change the function

In the two new prompts, sine is replaced by cosine and tangent. The list of integrals from the cosine functions are similar to those for the sine functions, except when the exponent is two. In the case of the tangent functions, the first and third integrals stand out as different.

In a structured inquiry, the teacher might allocate one of the two prompts to individual students or use one prompt after the other to initiate separate class-wide inquiries. Whichever approach the teacher takes, students should be inquiring more independently than they did on the original prompt.

2. Combine functions

Another line of inquiry comes from the Discovery Project: Patterns in Integrals. Question 2 looks at integrands in the form sin(ax)cos(bx) through a teacher-led investigation. Students find the integrals for particular cases and then generalise. The video (external link) shows the approach and gives the solutions to the teacher's questions.

The inquiry prompt (pictured above) shows two cases. While they are sufficient to suggest a general pattern, the prompt is open enough to encourage questions, conjectures, and exploration. What happens if we change the integrand to: