11.18 Applications of the differentiation of trigonometric functions
Tangents and normals to trigonometric functions
We can use our knowledge of differentiating trigonometric functions to find the equation of tangents and normals.
| $\frac{d}{dx}\sin f(x)$ddxsinf(x) | $=$= | $f'(x)\cos f(x)$f′(x)cosf(x) |
| $\frac{d}{dx}\cos f(x)$ddxcosf(x) | $=$= | $-f'(x)\sin f(x)$−f′(x)sinf(x) |
| $\frac{d}{dx}\tan f(x)$ddxtanf(x) | $=$= | $f'(x)\sec^2f(x)$f′(x)sec2f(x) |
When finding equations of tangents and normals, remember that the gradient of the tangent at any point on a function is given by the first derivative of the function. The following formulas are also useful:
Given the gradient $m$m and a point $(x_1,y_1)$(x1,y1), the equation of a line can be found using the point gradient formula:
$y-y_1=m(x-x_1)$y−y1=m(x−x1)
Given the gradient of the tangent $m_1$m1 at a point, the gradient of the normal $m_2$m2 is:
$m_2=-\frac{1}{m_1}$m2=−1m1
Practice question
Question 1
The curve $y=-5x\sin x$y=−5xsinx passes through the point $P\left(\pi,0\right)$P(π,0).
Find the equation of the tangent at point $P$P.
Determining the characteristics of, and sketching trigonometric functions
The first and second derivatives of trigonometric functions can be used to determine concavity, and regions where a trigonometric function is increasing or decreasing. Additionally, any stationary points and points of inflection can be found. All of these features will then help us to sketch the corresponding trigonometric functions.
Practice questions
Question 2
Consider the function $y=\frac{\sin x}{1+\cos x}$y=sinx1+cosx.
Find $\frac{dy}{dx}$dydx.
How many turning points does $y$y have?
Question 3
For the function $f\left(x\right)=x+\sin x$f(x)=x+sinx:
Find $f'\left(x\right)$f′(x)
Which correctly describes the behaviour of the function?
It is increasing for some values of $x$x but decreasing for others.
AIt is monotonically increasing.
BIt is never increasing for any value of $x$x.
CIt is monotonically decreasing.
DIt is never decreasing for any value of $x$x.
EIt is stationary for all values of $x$x.
FDetermine the $x$x-coordinate of any stationary points for $0\le x\le2\pi$0≤x≤2π.
Find $f''\left(x\right)$f′′(x).
Fill in the table below:
$x$x $0$0 $\frac{\pi}{2}$π2 $\pi$π $\frac{3\pi}{2}$3π2 $2\pi$2π $f'\left(x\right)$f′(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $f''\left(x\right)$f′′(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Hence choose the graph which represents $f\left(x\right)$f(x).
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Other applications
Oscillations are often modelled using trigonometric functions. Suppose the horizontal position of an object relative to a central point varies over time in a way that can be modelled using a sine or cosine function. We may wish to find an expression for the instantaneous velocity of the object at time $t$t, and then find features such as the times $t>0$t>0 when the velocity is at a maximum in magnitude. Due to the oscillatory nature of trigonometric functions, there will be infinitely many times at which a maximum or a minimum is reached, unless a finite domain is specified.
Practice question
Question 4
The arm of a pendulum swings between its two extreme points $A$A to the left and $B$B to the right. Its horizontal displacement $x$x cm from the centre of the swing at time $t$t seconds after it starts swinging is given by $x\left(t\right)=16\sin3\pi t$x(t)=16sin3πt.
At what position does the pendulum start swinging?
What is the furthest it gets from the central position of its swing?
Velocity is the rate at which displacement changes over time.
State the velocity function $v\left(t\right)$v(t).
Solve for the first two times at which the pendulum comes to rest.
What is the displacement of the pendulum when it first comes to rest?
What is the displacement of the pendulum when it comes to rest the second time?