Derivatives of sine and cosine

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curves are $y = \sin x$ and $y = \cos x$ ; the blue line is the tangent to $\sin x$ at $x = a$ . Move $a$ and compare the slope of the line with the height of the cosine curve at $x = a$ .

The sine curve, the cosine curve and the tangent to the sine curve at x = a. The tangent's slope is always the height of the cosine curve at a.

At $a = \pi/2 \approx 1.57$ , sine is at its top. What is the slope there, and what is $\cos(\pi/2)$ ?

Both are 0
Both are 1

Key idea

With $x$ in radians:

(\sin x)' = \cos x

and

(\cos x)' = -\sin x

.

Worked example

$f(x) = 3\sin x - 4\cos x$ . Find $f'(x)$ .

(3\sin x)' = 3\cos x

(sin x)' = cos x.

(-4\cos x)' = 4\sin x

(cos x)' = −sin x, and − · − = +.

f'(x) = 3\cos x + 4\sin x

Combine.

Your turn, guided

Your turn: $f(x) = 2\cos x + x^2$ .

f'(x) = -2\sin x + 2x

−2 sin x and 2x.

Practice ★

$f(x) = 3\sin x + 5\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 7\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 9\cos x$ . Find $f'(x)$ .

Watch out!

Practice ★★

Find the slope of the tangent line to $y = \sin x$ at $x = \dfrac{\pi}{ 2 }$ . Give an exact value.
Find the slope of the tangent line to $y = \sin x$ at $x = \dfrac{\pi}{ 3 }$ . Give an exact value.

$f(x) = 4x - 4\cos x$ . Find $f'(x)$ .
$f(x) = 4x - 8\cos x$ . Find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 3\sin x + 5\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 7\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 9\cos x$ . Find $f'(x)$ .

Summary card

Sine and cosine

$(\sin x)' = \cos x$ .
$(\cos x)' = -\sin x$ .

Watch out: the minus sign goes with the derivative of cosine; radians only.

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