Grade 11

Trigonometry

Extend trig beyond right triangles — sketch sin and cos graphs, use reduction formulae and solve equations in given intervals.

Sine and Cosine Graphs

The sine and cosine functions produce waves. You can stretch them vertically (amplitude), stretch them horizontally (period) and shift them up or down.

Theorem

y = a\sin(bx) + q \qquad y = a\cos(bx) + q

Amplitude = $|a|$ — half the distance from min to max
Period = $\dfrac{360°}{b}$ — how long one full cycle takes
Shift up by $q$ — max value = $|a| + q$ , min value = $-|a| + q$
$a<0$ flips the graph upside down

Interactive sin/cos graph

Try it yourself

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Worked Example

For $y = 3\sin(2x) - 1$ , state the amplitude, period, max value and min value.

The period formula is always

360°/b

for degrees. Check that you use the coefficient of

x

(not the whole angle expression) for

b