← Maths Extension 1 syllabus

NSWMaths Extension 1

Trigonometric Functions (ME-T1, T2, T3)

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we write $a \sin x + b \cos x$ as a single sinusoid, and what is this used for?

Express $a \sin x + b \cos x$ in the form $R \sin(x + \alpha)$ or $R \cos(x - \alpha)$ and use this to solve equations and find extreme values
A focused answer to the HSC Maths Extension 1 dot point on the auxiliary angle technique. Writing $a \sin x + b \cos x$ as a single sinusoid, finding the amplitude and phase, and using the result to solve equations and identify maxima and minima.
7 min answer →

How do we find every solution to a trigonometric equation, not just the principal one?

Write general solutions to trigonometric equations using the period and the symmetries of $\sin$ , $\cos$ and $\tan$
A focused answer to the HSC Maths Extension 1 dot point on general solutions of trigonometric equations. The general-solution formulas for $\sin \theta = k$ , $\cos \theta = k$ and $\tan \theta = k$ , restriction to given intervals, and equations with composite arguments.
7 min answer →

What are the inverse trigonometric functions, and what are their domains, ranges and graphs?

Define and sketch the inverse trigonometric functions $\arcsin$ , $\arccos$ and $\arctan$ , including their domains and ranges
A focused answer to the HSC Maths Extension 1 dot point on inverse trigonometric functions. Restricted domains for $\sin$ , $\cos$ and $\tan$ to define $\arcsin$ , $\arccos$ and $\arctan$ , their graphs, exact values, and identities, with worked examples.
9 min answer →

How do we convert between products and sums of trigonometric functions?

Use the product-to-sum and sum-to-product identities to simplify trigonometric expressions and integrals
A focused answer to the HSC Maths Extension 1 dot point on product-to-sum and sum-to-product identities. The four product-to-sum formulas, their sum-to-product converses, derivation from sum and difference, and use in integration, with worked examples.
7 min answer →

How do we expand $\sin(A \pm B)$ , $\cos(A \pm B)$ and $\tan(A \pm B)$ , and what are these identities used for?

Use the sum and difference identities for sine, cosine and tangent to expand or simplify trigonometric expressions
A focused answer to the HSC Maths Extension 1 dot point on sum and difference identities. The expansions of $\sin(A \pm B)$ , $\cos(A \pm B)$ and $\tan(A \pm B)$ , derivation of double-angle and half-angle formulas, and exact values for non-standard angles, with worked examples.
8 min answer →

How does the t-substitution $t = \tan \frac{\theta}{2}$ help simplify and solve trigonometric equations?

Use the t-formula (Weierstrass substitution) to express $\sin \theta$ , $\cos \theta$ and $\tan \theta$ as rational functions of $t = \tan \frac{\theta}{2}$
A focused answer to the HSC Maths Extension 1 dot point on the t-formula. Derivation of the t-substitution, the rational expressions for $\sin$ , $\cos$ and $\tan$ in terms of $t$ , and its use in solving and simplifying trig equations, with worked examples.
7 min answer →