Ch.6 - Trigonometric Functions - Reciprocal and Inverse Trigonometric Functions

Oct 13, 2026

Chapter 6 introduces reciprocal and inverse trigonometric functions for Edexcel Pure Year 2. It defines three reciprocal functions in terms of division by the original trig functions, explains where each is undefined due to zero denominators, and highlights common notation points such as inverse functions versus reciprocal powers. The chapter covers graph sketching and key identities, then shows how to simplify and prove reciprocal-function identities. It also explains inverse function definition via restricted domains, reflection in y = x, domain and range for inverse graphs, and solving equations that involve reciprocal and inverse forms.

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Reciprocal trigonometric functions

Definitions and domains

Graphing reciprocal functions

Identities and simplifying expressions

Proving identities

Solving equations with reciprocal functions

Inverse trigonometric functions

Inverse via restricted domains and reflection

Domain and range of inverse graphs

Sketching transformations and equations involving inverses

Trigonometric functions Cheat Sheet Edexcel Pure Year 2 Reciprocal trigonometric functions Previously, you have met three trigonometric functions; 􀝏􀝅􀝊􀝔 , 􀜿􀝋􀝏􀝔 􀜽􀝊􀝀 􀝐􀜽􀝊􀝔 . This chapter introduces three more trigonometric functions, known as the reciprocal trigonometric functions: ▪ 􀝏􀝁􀜿􀝔 = 􀯖􀯢1􀯦􀯫 (undefined for values of x for which 􀜿􀝋􀝏􀝔 = 0) ▪ 􀜿􀝋􀝏􀝁􀜿􀝔 = 􀯦􀯜1􀯡􀯫 (undefined for values of x for which 􀝏􀝅􀝊􀝔 = 0) ▪ 􀜿􀝋􀝐􀝔 = 􀯧􀯔1􀯡􀯫 (undefined for values of x for which 􀝐􀜽􀝊􀝔 = 0) Since division by zero is undefined, we have that these functions are undefined when the denominators are equal to zero. Note that 􀜿􀝋􀝐􀝔 = 􀯧􀯔1􀯡􀯫 = 􀯖􀯦􀯢􀯜􀯦􀯡􀯫􀯫 , simply by replacing 􀝐􀜽􀝊􀝔 with 􀯦􀯖􀯜􀯢􀯡􀯦􀯫􀯫 . This will sometimes be a more useful form to use. Careful: It is important to note that 􀜿􀝋􀝏 −1 (􀝔) does not equal sec(􀝔). In trigonometry, 􀜿􀝋􀝏−1 (􀝔) denotes the inverse function, 􀜽􀝎􀜿􀜿􀝋􀝏(􀝔) . On the other hand, sec(􀝔) = (cos(􀝔))−1, where the negative exponent indicates the reciprocal function. Similarly, we have 􀝏􀝅􀝊−1 (􀝔) = arcsin(􀝔), and (sin(􀝔))−1 = 􀜿􀝋􀝏􀝁􀜿 (􀝔) ;􀝐􀜽􀝊−1 (􀝔) = arctan(􀝔), and (tan(􀝔))−1 = 􀜿􀝋􀝐(􀝔) Graphing the reciprocal functions You need to be able to sketch the reciprocal trigonometric functions as well as any transformations, using radians and degrees. Below are the graphs of the reciprocal functions 􀝕 = sec 􀝔 Dotted curve represents 􀝕 = 􀜿􀝋􀝏􀝔 Domain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝋􀝀􀝀 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀰗2 (90°) Range: 􀝕 ≤ −1 , 􀝕 ≥ 1 􀝕 = 􀜿􀝋􀝏􀝁􀜿􀝔 Dotted curve represents 􀝕 = 􀝏􀝅􀝊􀝔 Domain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀟨 (180°) Range: 􀝕 ≤ −1 􀝋􀝎 􀝕 ≥ 1 􀝕 = 􀜿􀝋􀝐􀝔 Domain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀟨 (180°) Range: 􀝕 ∈ ℝ This gives us the following identities: ▪ 1 + 􀜿􀝋􀝐 2􀝔 = 􀜿􀝋􀝏􀝁􀜿 2􀝔 ▪ 1 + 􀝐􀜽􀝊2􀝔 = 􀝏􀝁􀜿 2􀝔 You could be asked to prove these identities, so make sure you are able to reproduce the proofs on the left. Simplifying expressions and proving identities You can use the definitions and identities we have covered so far to simplify and prove expressions involving the reciprocal trig functions. There is no trick or standard procedure to be used for these questions. Your ability to manipulate trigonometric expressions using reciprocal functions and identities is being tested, so the most useful thing you can do is properly familiarise yourself with these functions and the above identities. As with most of mathematics, the most useful tool here is practice. When proving identities, you must start from one side and work your way towards the other side. You can start from any side, so pick whichever seems like an easier starting point. | Example 1: Prove that 􀝏􀝁􀜿 2􀝔 + 􀜿􀝋􀝏􀝁􀜿 2􀝔 = 􀝏􀝁􀜿 2􀝔 􀜿􀝋􀝏􀝁􀜿 2􀝔 | | | | | | --- | --- | --- | --- | --- | | Starting from the LHS, we have:
using the 􀝐􀜽􀝊2􀝔 and 􀝏􀝁􀜿 2􀝔 identities:
Rewriting 􀝐􀜽􀝊􀝔 as 􀯦􀯖􀯜􀯢􀯡􀯦􀯫􀯫 and 􀜿􀝋􀝐􀝔 as 􀯖􀯦􀯢􀯜􀯦􀯡􀯫􀯫 :
combining everything into one fraction:
using 􀝏􀝅􀝊2􀝔 + 􀜿􀝋􀝏 2􀝔 = 1
splitting the fraction up into a product, giving us the RHS | 􀜮􀜪􀜵 = 􀝏􀝁􀜿 2􀝔 + 􀜿􀝋􀝏􀝁􀜿 2􀝔
(1 + 􀝐􀜽􀝊2􀝔) + (1 + 􀜿􀝋􀝐 2􀝔) = 2 + 􀝐􀜽􀝊2􀝔 + 􀜿􀝋􀝐 2􀝔 | | | | | | = 2 + | 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 | + | 􀜿􀝋􀝏 2􀝔􀝏􀝅􀝊2􀝔 | | | 􀝏􀝅􀝊4􀝔 + 􀜿􀝋􀝏 4􀝔= 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 + 2
􀝏􀝅􀝊4􀝔 + 􀜿􀝋􀝏 4􀝔 + 2􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔
=
􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔(􀝏􀝅􀝊2􀝔 + 􀜿􀝋􀝏 2􀝔)2 1
= =
􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔
1 1
= 􀝏􀝅􀝊2􀝔 ∙ 􀜿􀝋􀝏 2􀝔 = 􀝏􀝁􀜿 2􀝔 􀜿􀝋􀝏􀝁􀜿 2􀝔 = 􀜴􀜪􀜵 | | | | Solving equations Previously, in Pure Year 1, you learnt how to solve trigonometric equations involving 􀝏􀝅􀝊􀝔, 􀜿􀝋􀝏􀝔 􀜽􀝊􀝀 􀝐􀜽􀝊􀝔 . Now we will look at solving equations that also involve the reciprocal functions. The only difference here is that you need to use the identities and definitions we have covered in this chapter in order to simplify the equation, before you can solve it. | Example 2: Solve the equation 􀝏􀝁􀜿􀝔 = ξ2 in the interval 0 ≤ 􀝔 ≤ 360° | | | | --- | --- | --- | | rewriting 􀝏􀝁􀜿􀝔 as 􀯖􀯢1􀯦􀯫 :
taking the reciprocal of both sides:
We can solve this via CAST or a graphical method giving: | 1
􀜿􀝋􀝏􀝔 | = ξ2 | | | ∴ 􀜿􀝋􀝏􀝔 = 1ξ2 􀝔 = 45°, 315° . | | | Example 3: Solve 􀝏􀝁􀜿 2􀝔 = 3􀝐􀜽􀝊􀝔 in the interval 0 ≤ 􀝔 ≤ 360

FAQs

What approach is recommended for proving identities involving reciprocal trig functions?

The chapter emphasizes that there is no single trick or standard procedure. It recommends starting from one side of the identity and working step-by-step toward the other side using the covered definitions and identities, supported by practice.

Oct 13, 2026

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