Trigonometry

10 methods · 13 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

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Before solving any trig equation, reduce it to one of the three canonical shapes and write every solution at once: $sin A = sin B \Rightarrow A = B + 2 k π$ or $A = π - B + 2 k π$ ; $cos A = cos B \Rightarrow A = \pm B + 2 k π$ ; $tan A = tan B \Rightarrow A = B + k π$ , then keep only the roots in range.

Trigger: any equation reducible to equality of two like trig ratios, especially when a full family or a sketch of all solutions is wanted.

Instances:
(i) $sin y = sin x$ gives the lines $y = x + 2 k π$ and $y = π - x + 2 k π$ .
(ii) $cos (x - α) = cos β$ gives $x - α = \pm β + 2 k π$ , hence $tan x = tan (α \pm β)$ .
(iii) A $tan$ equation needs the $+ k π$ period, not $2 k π$ , or solutions are lost.

Linked questions (3)

STEP 2 2008 · Q6

A curve has the equation

y = f (x)

, where

f (x) = cos (2 x + \frac{π}{3}) + sin (\frac{3 x}{2} - \frac{π}{4}) .

Find the period of $f (x)$ .
Determine all values of $x$ in the interval $- π ⩽ x ⩽ π$ for which $f (x) = 0$ . Find a value of $x$ in this interval at which the curve touches the $x$ -axis without crossing it.
Find the value or values of $x$ in the interval $0 ⩽ x ⩽ 2 π$ for which $f (x) = 2$ .

STEP 2 2011 · Q4

Find all the values of $θ$ , in the range $0^{\circ} < θ < 18 0^{\circ}$ , for which $cos θ = sin 4 θ$ . Hence show that $sin 1 8^{\circ} = \frac{1}{4} (5 - 1) .$
Given that $4 sin^{2} x + 1 = 4 sin^{2} 2 x,$ find all possible values of $sin x$ , giving your answers in the form $p + q 5$ where $p$ and $q$ are rational numbers.
Hence find two values of $α$ with $0^{\circ} < α < 9 0^{\circ}$ for which $sin^{2} 3 α + sin^{2} 5 α = sin^{2} 6 α .$

STEP 2 2017 · Q3

Sketch, on $x$ - $y$ axes, the set of all points satisfying $sin y = sin x$ , for $- π ⩽ x ⩽ π$ and $- π ⩽ y ⩽ π$ . You should give the equations of all the lines on your sketch.
Given that $sin y = \frac{1}{2} sin x$ obtain an expression, in terms of $x$ , for $y^{'}$ when $0 ⩽ x ⩽ \frac{1}{2} π$ and $0 ⩽ y ⩽ \frac{1}{2} π$ , and show that $y^{''} = - \frac{3 sin x}{( 4 - sin ^{2} x ) ^{\frac{3}{2}}} .$ Use these results to sketch the set of all points satisfying $sin y = \frac{1}{2} sin x$ for $0 ⩽ x ⩽ \frac{1}{2} π$ and $0 ⩽ y ⩽ \frac{1}{2} π$ . Hence sketch the set of all points satisfying $sin y = \frac{1}{2} sin x$ for $- π ⩽ x ⩽ π$ and $- π ⩽ y ⩽ π$ .
Without further calculation, sketch the set of all points satisfying $cos y = \frac{1}{2} sin x$ for $- π ⩽ x ⩽ π$ and $- π ⩽ y ⩽ π$ .