Derivatives of sine/cosine/tangent | Year 12

01 / LEARN

The idea, step by step

Radians are part of the rule

When the input is in radians, ${(sin x)}^{'} = cos x$ , ${(cos x)}^{'} = - sin x$ and ${(tan x)}^{'} = 1 / {cos}^{2} x = {sec}^{2} x$ (where $sec x = 1 / cos x$ ). These formulas do not apply unchanged to numerical degree inputs.

For a scaled angle, multiply by its derivative. The derivative of $sin (3 x)$ is $3 cos (3 x)$ , not $cos x$ .

Structure and domain

Use product and chain rules when trigonometric expressions are combined. Squared sine means ${(sin x)}^{2}$ , so its derivative uses the chain rule.

The derivative of tangent is defined only where tangent itself is defined: exclude $x = π / 2 + k π$ for integers $k$ , with adjustments for any scaled angle.

A common trap

Losing the negative sign when differentiating cosine.

Instead: Cosine decreases immediately to the right of zero; its derivative is $- sin x$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Scaled trigonometric angles

Differentiate $y = 3 cos (2 x) - sin x$ .

The derivative of $3 cos (2 x)$ is $- 6 sin (2 x)$ .
The derivative of $- sin x$ is $- cos x$ .

- 6 sin (2 x) - cos x

Example 2

A periodic height model

A height is $h (t) = 2 + sin (π t / 4)$ metres. Find its rate at $t = 0$ seconds.

$h^{'} (t) = (π / 4) cos (π t / 4)$ .
At zero, the cosine is 1.

π / 4

metres per second

Check your understanding

Differentiate ${sin}^{2} x$ .

Show answer

$2 sin x cos x$

Apply the power rule to the outer square and multiply by the derivative of sine.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Differentiate $5 sin x + 2 cos x$ .
2 marks
Differentiate $tan (3 x)$ .
2 marks
Find the gradient of $y = cos (2 x)$ at $x = π / 4$ .
2 marks
Differentiate $x sin x$ .
2 marks
Differentiate ${cos}^{3} x$ .
2 marks
For $h (t) = 5 + 2 cos (π t / 6)$ metres, find and interpret $h^{'} (3)$ in m/s.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate $5 sin x + 2 cos x$ .
2 marks
Apply the sine and cosine rules term by term.
$5 cos x - 2 sin x$
Differentiate $tan (3 x)$ .
2 marks
Use the tangent rule.
Multiply by 3 from the inner angle.
$3 {sec}^{2} (3 x)$
Find the gradient of $y = cos (2 x)$ at $x = π / 4$ .
2 marks
The derivative is $- 2 sin (2 x)$ .
At $π / 4$ , the sine is 1.
$- 2$
Differentiate $x sin x$ .
2 marks
Use the product rule.
Differentiate one factor at a time.
$sin x + x cos x$
Differentiate ${cos}^{3} x$ .
2 marks
Differentiate the outer cube.
Multiply by $- sin x$ .
$- 3 {cos}^{2} x sin x$
For $h (t) = 5 + 2 cos (π t / 6)$ metres, find and interpret $h^{'} (3)$ in m/s.
3 marks
Differentiate to get $- (π / 3) sin (π t / 6)$ .
At $t = 3$ , the angle is $π / 2$ .
$- π / 3$ m/s: the height is decreasing.

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Curriculum reference & lesson scope

SACE Stage 2 Mathematical Methods

Further differentiation and applications from the current SACE Methods outline. Logarithmic derivatives have their own chapter; integration follows later. Trigonometric calculus uses radians.

Official curriculum source ↗

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