Trigonometric Functions (ME-T1, T2, T3)

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 by adding two waves into one, finding the amplitude $R$ and phase $\alpha$ from a right triangle, and using the result to solve equations and identify maxima and minima, with stage-by-stage diagrams and worked examples.

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 $\tan \theta = k$ read off the unit circle, restriction to a given interval shown on a number line, and equations with composite arguments, with stage-by-stage diagrams and worked examples.

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 $\arctan$, the stage-by-stage construction of the $\arcsin$ graph by reflection, their domains, ranges, exact values and identities, with worked examples.

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 the sum and difference identities, the wave-superposition picture, and use in integration and equation solving, with worked examples.

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 $\tan(A \pm B)$ seen on the unit circle, derivation of the double-angle and half-angle formulas, and exact values for non-standard angles, with worked examples.

A focused answer to the HSC Maths Extension 1 dot point on the t-formula. The half-angle right triangle, 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 and integrals, with worked examples.