Mathematics (2 unit) – Trigonometric Functions

The Trigonometric Functions

To study the calculus relating to the trigonometric functions, we must firstly study the radian as a measure of angle. The radian as a unit for angle measure is more suitable when considering the trigonometric functions. We shall now study this measure of angle.

Definition: A radian is the size of the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle. This situation is depicted on the diagram below.

Now, to measure the number of radians in one revolution $Latex formula$ we simply measure the number of radius lengths within the circumference of the circle. To do so we have,

$Latex formula$

as the number of radius lengths within the circumference of the circle.

Hence we have that the number of radians within a revolution is $Latex formula$. Hence we have that,

$Latex formula$

Hence we have that,

 $Latex formula$

The following common angle conversions should be known by heart.

 Radians $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ Degrees $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$

To convert other angles, we simply form an equation using ratios. It should be obvious that the ratio of any angle in radians over $Latex formula$ is equal to the ratio of that angle in degrees over $Latex formula$. From this, the unknown is left as some pronumerals and the resultant equation is solved to obtain the required value. Consider the below example to illustrate the conversion of angles.

Example 1

a) Convert $Latex formula$ into a radian measure.

b) Convert $Latex formula$ into degree measure.

Solution 1

a) We simply create an equation using the equality of ratios as explained above.

$Latex formula$

Hence we have that $Latex formula$ radians.

b) Again, we simply create a ratio to find the angle in question.

$Latex formula$

Hence we have that $Latex formula$

Note: When degrees are intended, the degree sign is included. For radians, at times the angle measure is followed by the word “rad”, but most of the time an angle in radians is left without any units following after.

Note: The reason for the use of the radian angle of measure is because functions such as $Latex formula$ do not make physical sense when the domain is in degrees. Also, the degree unit of measure is based on the angle subtended by the Earth at the sun after one day (the sun is the centre of the circle made by the Earth’s orbit). Obviously mathematics is universal and is not to be restricted only to our solar system, hence making radians a more universal angle measure and more suitable for mathematics.

Arc length and sector area

Within a circle of radius $Latex formula$ we have that the following formulae are true.

 $Latex formula$$Latex formula$

where $Latex formula$ is measured in radians.

The first formula gives the length of an arc. The second formula gives the area of the sector of the circle. The most important aspect here is that the angle must be measured in radians.

Now at this point, we can extrapolate the second formula to obtain a formula for the area of the minor segment subtended by the angle. Consider the shaded region below which is the minor segment subtended by the angle at the centre.

Now, obviously the area of this minor segment is equal to the area of the sector subtended by the angle $Latex formula$ minus the area of the isosceles triangle subtended within the sector. That is,

$Latex formula$

Since we have that the isosceles triangle has two sides, each of length  about an angle of size $Latex formula$ radians. Hence we have that the area of the minor segment is given by the formula:

 $Latex formula$

where $Latex formula$ is measured in radians. Be sure to convert your calculator into radian mode to ensure that the correct value of $Latex formula$ is obtained.

Note: Ensure you understand how to convert your calculator from the degree mode of angle measure to the radian mode of angle measure. Use the latter case when calculating trigonometric ratios of angles in radians.

Example 2

An arc of a circle with 8 cm subtends an angle of $Latex formula$ at the centre. Find the length of the arc and the area of the sector.

Solution 2

So, to find the length of the arc and the area of the sector, we firstly must convert degrees into radians. We have that $Latex formula$ converts to $Latex formula$ radians. Now, using the formula for arc length we have,

$Latex formula$

To find the area of the sector we simply substitute into the formula.

$Latex formula$

$Latex formula$

Do not forget to add the appropriate units.

Example 3

In a circle with radius $Latex formula$m, find the area of the minor segment subtended by an angle of $Latex formula$ radians at the centre.

Solution 3

To find the area of the minor segment, we simply use the formula derived earlier.

$Latex formula$

Now, we have that $Latex formula$m and $Latex formula$ radians. Substituting into the formula gives,

$Latex formula$

$Latex formula$

Example 4

A sector of a circle has an area of $Latex formula$. If the angle at the centre is $Latex formula$ radians, find the radius of the circle correct to two significant figures.

Solution 4

We simply use the formula for the area of a sector to calculate the radius.

$Latex formula$

Now, substituting the respective values gives,

$Latex formula$

$Latex formula$

Hence the radius of the circle is $Latex formula$ mm correct to two significant figures.

Graphs of the basic Trigonometric Functions

You should be familiar with the graphs of the trigonometric functions $Latex formula$ with the measure of the angle being the degree. We shall now investigate these graphs, using radians as our angle measure. Recall that the trigonometric functions are periodic, which implies that for a function $Latex formula$ there exists $Latex formula$ such that $Latex formula$. We firstly note that the period of a function is the value of $Latex formula$ in the above identity. That is, it is simply the difference between functional values that results in the functional values repeating themselves. Also, the amplitude of a periodic function is the maximum value of  for all $Latex formula$ within the domain. Below every curve is a description of the most important facts relating to it.

$Latex formula$

This curve has zeros at the values $Latex formula$ where $Latex formula$ is an integer. The period of revolution is $Latex formula$. The amplitude of the curve is $Latex formula$.

$Latex formula$

This curve has zeros at $Latex formula$ where $Latex formula$ is an integer. The period of revolution is $Latex formula$. The amplitude of the curve is 1.

$Latex formula$

The curve has zeros at $Latex formula$ where $Latex formula$ is an integer. The curve has asymptotes at $Latex formula$ where $Latex formula$ is an integer. The period of the function is $Latex formula$ radians.

$Latex formula$

The curve has no zeros. The period of the curve is $Latex formula$. The curve has asymptotes at $Latex formula$ where $Latex formula$ is an integer.

$Latex formula$

The curve has no zeros. The period of the curve is $Latex formula$. Asymptotes are at $Latex formula$ where $Latex formula$ is an integer.

$Latex formula$

The zeros of the curve occur at $Latex formula$ where $Latex formula$ is an integer. The asymptotes occur at $Latex formula$ where $Latex formula$ is integral. The period of the function is $Latex formula$.

More difficult graphs

The Advanced Mathematics syllabus may also ask students to sketch variations (translations and stretches) of the two main trigonometric graphs; $Latex formula$ and $Latex formula$.

In general for the curve $Latex formula$, the point $Latex formula$ is translated to the point $Latex formula$. The amplitude of the curve is $Latex formula$ and the period of the curve is $Latex formula$.

In general for the curve $Latex formula$, the point $Latex formula$ is translated to the point $Latex formula$ and the origin is converted to the point $Latex formula$. As before the amplitude of the curve is $Latex formula$ and the period curve is $Latex formula$.

These facts should be memorised. There are also other methods of sketching trigonometric curves (such as plotting points) which may complement the methods taught here.

Note: Almost all questions involving sketching will indicate the domain to be sketched.

Consider the below examples that show utilisation of these formulae.

Example 5

Sketch the graph of $Latex formula$ for $Latex formula$.

Solution 5

Here we firstly notice that the domain we are required to sketch the curve for is $Latex formula$.

Now, the origin is translated to the point $Latex formula$. The amplitude of the curve is $Latex formula$ and so the maximum and the minimum values of the function are $Latex formula$ and $Latex formula$ respectively. The period of the curve is $Latex formula$ and hence the curve repeats itself every $Latex formula$ radians. Since the four phases of the graph (equilibrium, minima, equilibrium, maxima) all occur within the period of $Latex formula$ radians, then it follows that we must consider the graph in intervals of $Latex formula$ from the translated origin. Also, as for any curve sketching question, we find the $Latex formula$-intercept. At $Latex formula$, $Latex formula$. Hence the $Latex formula$-intercept is $Latex formula$. Also, to find the $Latex formula$-intercepts, we can simply solve the equation  for $Latex formula$. However, since we previously found the minima of the function to be zero, it thus follows that the intercepts will show themselves once the curve has been plotted. So, using the above information and extrapolating the curve gives the diagram,

Which is the graph of $Latex formula$ for $Latex formula$ lieing between $Latex formula$ and $Latex formula$.

Note: It is very easy to become confused when solving questions involving trigonometric functions. To help with the sketching of the curve, plotting a few easy points helps. In the above case, noting the $Latex formula$-value at $Latex formula$ and $Latex formula$ will help with sketching the curve.

The next example illustrates the use of the formulae related to the cosine function.

Example 6

Sketch the graph of $Latex formula$ for $Latex formula$.

Solution 6

Firstly, we notice that the point $Latex formula$ is translated to the point $Latex formula$. Secondly, the period of the graph is $Latex formula$ and the amplitude of the graph is $Latex formula$. The equilibrium position is the line $Latex formula$. Finding the exact $Latex formula$-intercepts is not important in this case since finding them will involve approximations.  Now, using this information to sketch the graph gives,

Differential Calculus involving the Trigonometric Functions

We shall now study the differential calculus of the trigonometric functions. We shall not show the first principles derivation of these differentials; rather we shall present them for the student to memorise. The reason for not showing a derivation is due to the requirement of several results outside the scope of the advanced mathematics course.

 $Latex formula$ $Latex formula$ $Latex formula$

The three results shown here must be memorised. The differentials of the other trigonometric functions may be obtained using the chain rule and quotient rules.

Applying the chain rule to the above differentials gives the following results.

 $Latex formula$ $Latex formula$ $Latex formula$

We shall now apply these results to some example questions.

Example 7

Differentiate the following functions with respect to $Latex formula$.

a) $Latex formula$

b) $Latex formula$

c) $Latex formula$

d) $Latex formula$

e) $Latex formula$

f) $Latex formula$

Solution 7

a) $Latex formula$ $Latex formula$

b) $Latex formula$ $Latex formula$

c) $Latex formula$ $Latex formula$

d) $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$

e) $Latex formula$ $Latex formula$

f) $Latex formula$ $Latex formula$

As exhibited in the above example, all the rules that previously applied for differentiation still apply for differentiation (i.e. the chain rule, the product rule, the quotient rule). You also must be able to use the derivative to find gradients and equations of tangents and normals and also maxima and minima.

Example 8

Find the equation of the tangent to $Latex formula$ at the point where $Latex formula$.

Solution 8

So firstly we differentiate the function to obtain the gradient at $Latex formula$.

$Latex formula$

At $Latex formula$, $Latex formula$. Hence we have the tangent at $Latex formula$ is horizontal.

Hence we have that the equation of the tangent is $Latex formula$. That is the equation is,

$Latex formula$

Example 9

Show that $Latex formula$ satisfies

$Latex formula$

Solution 9

So we simply differentiate the expression twice and show that it satisfies the relation.

$Latex formula$

$Latex formula$

Now, looking at the LHS of the above expression;

$Latex formula$

Hence we have that the function $Latex formula$ satisfies the required relation.

Integral Calculus involving the Trigonometric Functions

As differentiation is the reverse process of integration, we can simply derive the integrals of the trigonometric functions by applying the fundamental theorem of calculus. The following results should be committed to memory.

 $Latex formula$ $Latex formula$ $Latex formula$

The following three rules are given in the HSC Mathematics Standard Integrals table. These do not have to be committed to memory.

 $Latex formula$ $Latex formula$ $Latex formula$

Now, the more general forms of these rules are;

 $Latex formula$ $Latex formula$ $Latex formula$

All formulae for finding volumes, areas and definite integrals apply to the trigonometric functions as well. We shall now investigate some examples of such questions.

Example 10

Find the integral of $Latex formula$.

Solution 10

We are required to find $Latex formula$

Now, since the argument of the sine function is a linear form, then it follows that we may use the rules provided in the Standard Integral table. This gives,

$Latex formula$

Of course, the answer should be checked by differentiating the RHS and ensuring that the argument of the integrand is obtained.

Example 11

Find the definite integral

$Latex formula$

Solution 11

To find this integral, we simply find an anti-derivative and then use the fundamental theorem of calculus as required.

$Latex formula$

$Latex formula$

Noting here that $Latex formula$.

Example 12

Find the area under the curve $Latex formula$ between $Latex formula$ and $Latex formula$.

Solution 12

Firstly, we give a rough sketch of the curve. The amplitude is $Latex formula$ and the period is $Latex formula$. Hence we have the curve;

The shaded region is the area required by the curve. Hence we have;

$Latex formula$

$Latex formula$

$Latex formula$

Example 13

Differentiate $Latex formula$ and hence find the value of

$Latex formula$

Solution 13

So firstly we have that,

$Latex formula$

After applying the chain rule. That is we have that,

$Latex formula$

Integrating both sides with respect to $Latex formula$ gives (applying the fundamental theorem of calculus),

$Latex formula$

That is,

$Latex formula$

Hence we can now evaluate the definite integral.

$Latex formula$

Example 14

a) Find the primitive of $Latex formula$ using the identity

$Latex formula$

b) Hence find the volume formed when the area underneath the curve $Latex formula$ between $Latex formula$ and $Latex formula$ is rotated about the $Latex formula$-axis, correct to 2 decimal places.

Solution 14

a) So, to find the primitive of $Latex formula$ we simply use the identity provided to re-write  in terms of $Latex formula$.

$Latex formula$

$Latex formula$

b) So, using the formula for volume,

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$