Trigonometric Functions (ME-T1, T2, T3)

6 dot points across 6 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

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

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.

16 min read →

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

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.

14 min read →

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

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.

15 min read →

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}

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.

14 min read →

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)

R \cos(x - \alpha)

and use this to solve equations and find extreme values

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.

15 min read →

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

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.

15 min read →