Trigonometry

9 methods · 35 worked questions using these methods

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For an equation in $sin (k x)$ or $cos (k x)$ asked over an interval (how many solutions, the largest, or the sum), first transform the range of the inner argument (if $0 \leq x \leq L$ then $0 \leq k x \leq k L$ ), solve the base equation across that widened interval, then map each value back by dividing by $k$ . The widened interval is where you actually count.

Trigger: a multiple angle $k x$ inside the trig function and a 'how many / largest / sum of solutions' question.

Instances:
(i) stretch $0 \leq x \leq 2 π$ to $0 \leq 2 x \leq 4 π$ (or $72 0^{\circ}$ ) and list the $2 x$ values before dividing;
(ii) stretch to $0 \leq 3 θ \leq 54 0^{\circ}$ for a triple angle and count crossings of $cos 3 θ = k$ ;
(iii) intersect the $k x$ -solutions of two conditions, keep the common ones, then halve;
(iv) for a wide outer interval, count how many full periods fit and add the partial-period solutions.

Linked questions for this method (3)

TMUA 2016 Paper 1 · Q8

TMUA 2018 Paper 1 · Q18

TMUA 2018 Paper 2 · Q4