Year 12: Trigonometric Functions

5 dot points across 5 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do amplitude, period, phase shift and vertical shift transform the graphs of sine, cosine and tangent?

Sketch and interpret graphs of

y = a \sin(b x + c) + d

y = a \cos(b x + c) + d

and

y = a \tan(b x + c) + d

, identifying amplitude, period, phase shift and vertical shift

Graphs of trigonometric functions. Key features of $\sin x$ , $\cos x$ and $\tan x$ , and how amplitude $a$ , period $\frac{2 \pi}{b}$ , phase shift and vertical shift transform them, built up stage by stage, with worked examples.

15 min read →

How are radians defined, and how do we use them to find arc length and sector area?

Use radian measure to find arc length, the area of a sector, and the area of a segment of a circle

Radians and circular measure. Definition of radian built up stage by stage, conversion between radians and degrees, exact values, arc length $\ell = r \theta$ , sector area $A = \frac{1}{2} r^2 \theta$ , and area of a segment, with worked examples.

14 min read →

How do we find all solutions of a trigonometric equation in a given interval, including equations involving multiple angles and identities?

Solve trigonometric equations over a given interval using exact values, the unit circle, and identities to reduce to a single trig function

Solving trig equations. Principal values, the unit circle and the ASTC quadrant rule, all solutions in an interval found stage by stage, multiple angle equations with interval expansion, equations using identities to reduce to a single function, and quadratics in $\sin$ or $\cos$ , with worked examples.

15 min read →

Which trigonometric identities are essential for simplifying expressions and proving equivalences in HSC Maths Advanced?

Use Pythagorean, ratio, double angle and complementary identities to simplify expressions and prove equalities

Trigonometric identities. The Pythagorean identity with its unit-circle origin, ratio identities, complementary angle identities, and the double angle formulas, with a stage-by-stage proof method and worked examples.

15 min read →

How do the sine, cosine and area rules solve non-right-angled triangles, and how are they applied to elevation, depression, bearings and three-dimensional problems?

Solve problems using the sine rule, cosine rule and area rule, including angles of elevation and depression, bearings and three-dimensional applications

Triangle trigonometry. The sine rule and its ambiguous SSA case, the cosine rule in both side and angle forms, the area rule, angles of elevation and depression, compass and true bearings, and three-dimensional applications, with stage-by-stage diagrams and worked HSC-style examples.

18 min read →