Applications of trigonometric derivatives

Tangents and normals to trigonometric curves, finding stationary points and sketching curves such as y = sin x + cos x and y = x + sin x, and optimisation problems involving angles.

Tangents and normals

Example 1●●

Find the tangent and normal to $y = \sin x$ at $x = \pi$.

The point is $(\pi, 0)$ and the gradient is $\cos\pi = -1$.

Tangent: $y = -(x - \pi)$, that is $y = \pi - x$. Normal (gradient $1$): $y = x - \pi$.

Stationary points and sketching

Example 2●●●

Find the stationary points of $y = \sin x + \cos x$ for $0 \le x \le 2\pi$.

$y' = \cos x - \sin x = 0$ gives $\tan x = 1$, so $x = \dfrac{\pi}{4}$ or $x = \dfrac{5\pi}{4}$.

Solve y′ = 0

$y\left(\tfrac{\pi}{4}\right) = \tfrac{1}{\sqrt2} + \tfrac{1}{\sqrt2} = \sqrt2$ and $y\left(\tfrac{5\pi}{4}\right) = -\sqrt2$.

Values

$y'' = -\sin x - \cos x = -y$. It is negative at $\tfrac{\pi}{4}$ (maximum) and positive at $\tfrac{5\pi}{4}$ (minimum).

Nature

y = sin x + cos x

The maximum is at x = π/4 ≈ 0.79 and the minimum at x = 5π/4 ≈ 3.93.

Example 3●●

Find the stationary point of $y = x + \sin x$ for $0 \le x \le 2\pi$.

$y' = 1 + \cos x \ge 0$, and it is zero only at $x = \pi$.

The gradient is positive on both sides, so $(\pi, \pi)$ is a horizontal point of inflection.

Optimisation with angles

Example 4●●

Two sides of a triangle are $10$ cm long, and the angle between them is $\theta$. Find the greatest possible area.

Area $A = \tfrac12 \times 10 \times 10\sin\theta = 50\sin\theta$, for $0 < \theta < \pi$.

$A' = 50\cos\theta = 0$ at $\theta = \dfrac{\pi}{2}$, and $A'' = -50\sin\theta < 0$, so it is a maximum.

The greatest area is $50$ cm², when the angle is a right angle.

Flashcards

Tangent to $y = sin x$ at the origin

$y = x$

Maximum of $sin x + cos x$

$\sqrt2$, at $x = \tfrac{\pi}{4}$

Area of a triangle from two sides

$\tfrac12 ab\sin C$

Practice

The tangent to $y = \sin x$ at $x = \pi$ is:
Find the gradient of $y = \sin x$ at $x = \dfrac{\pi}{3}$.
Find the gradient of $y = \cos x$ at $x = \dfrac{\pi}{2}$.
On $0 \le x \le 2\pi$, $y = \sin x + \cos x$ has a maximum at:
Find the maximum value of $y = \sin x + \cos x$. Type $\sqrt2$ as sqrt2.
The stationary point of $y = x + \sin x$ at $x = \pi$ is a:
Two sides of a triangle are $10$ cm. Find the greatest possible area in cm².
The tangent to $y = \cos x$ at $x = \dfrac{\pi}{2}$ cuts the $y$-axis at:
Find the gradient of $y = \tan x$ at $x = 0$.