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This scheme of work for Year 13 trigonometric functions builds on sine, cosine and tangent by adding their reciprocals and their inverses. It develops the skill of matching each new function to the sine, cosine or tangent it comes from. These ideas are used when an equation or an identity involves secant, cosecant or cotangent, and when an output has to be an angle in a fixed range.
Students define and sketch the reciprocal functions, then solve equations with them. They prove the two identities that come from sine squared plus cosine squared equals 1. They finish by sketching inverse sine, inverse cosine and inverse tangent.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
Secant is 1 divided by cosine, cosecant by sine, and cotangent by tangent. Each is undefined where the denominator is zero. Secant and cosecant never take a value strictly between -1 and 1.
Divide sine squared plus cosine squared equals 1 by cosine squared to get 1 plus tan squared equals secant squared. Divide by sine squared to get 1 plus cot squared equals cosecant squared. Do not swap those two results.
Inverse sine is the angle whose sine is the input, not 1 divided by sine. Its outputs run from -pi/2 to pi/2, and its inputs from -1 to 1. Inverse cosine outputs from 0 to pi. Inverse sine of sine theta equals theta only inside that output range.
Reflect the allowed part of the graph in the line y = x. Inverse tangent has horizontal asymptotes at pi/2 and -pi/2. For tangent of inverse cosine of x, name the angle, then use a right-angled triangle.
| Misconception | Teaching focus |
|---|---|
| Reading secant as 1 divided by sine, or cotangent as 1 divided by sine. | Secant goes with cosine, cosecant with sine, and cotangent with tangent. The asymptote is where that pair is zero. |
| Writing 1 plus tan squared equals cosecant squared, or 1 plus cot squared equals secant squared. | Dividing by cosine squared gives secant. Dividing by sine squared gives cosecant. Match the function to the divisor. |
| Solving secant theta equals 1/2, or treating an asymptote as a solution. | Secant cannot lie strictly between -1 and 1. If no cosine fits, there is no solution. An asymptote is not a root. |
| Treating inverse sine as 1 divided by sine. | Inverse sine is the angle. The reciprocal of sine is cosecant. Say inverse sine, not sine to the power -1. |
| Giving an inverse sine output outside -pi/2 to pi/2, or an inverse cosine output outside 0 to pi. | The range is part of the definition. A value outside that range is not the output of the inverse function. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all four lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn how to work with reciprocal trigonometric functions: secant, cosecant, and cotangent. They begin by calculating exact values and interpreting their definitions in terms of sine, cosine, and tangent.
As learning progresses, they sketch graphs and solve equations involving these functions.
Students learn how to simplify expressions involving inverse trigonometric functions.
Later, as learning progresses, they solve trigonometric equations identities involving secant, cosecant and cotangent.
Students learn how to prove and apply identities involving secant, cosecant, and cotangent, starting with fundamental identities like 1 + cot^2 x = csc^2 x. They manipulate expressions using reciprocal and Pythagorean identities.
As learning progresses, they tackle more advanced proofs and apply identities to solve equations, justify equalities, and substitute into compound expressions.
Students learn how to sketch the functions arcsin, arccos and arctan.
Later, as learning progresses, they use graphs of arcsin, arccos and arctan to evaluate expressions such as tan(arccos x).
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