Now that we have graphs for
$sin\theta $ and
$cos\theta $ , there is an easy way to visualise the tangent graph. Let us look back at our definitions of
$sin\theta $ and
$cos\theta $ for a right-angled triangle.
This is the first of an important set of equations called
trigonometric identities . An identity is an equation, which holds true for any value which is put into it. In this case we have shown that
$$tan\theta =\frac{sin\theta}{cos\theta}$$
for any value of
$\theta $ .
So we know that for values of
$\theta $ for which
$sin\theta =0$ , we must also have
$tan\theta =0$ . Also, if
$cos\theta =0$ our value of
$tan\theta $ is undefined as we cannot divide by 0. The graph is shown in
[link] . The dashed vertical lines are at the values of
$\theta $ where
$tan\theta $ is not defined.
Functions of the form
$y=atan\left(x\right)+q$
In the figure below is an example of a function of the form
$y=atan\left(x\right)+q$ .
Functions of the form
$y=atan\left(\theta \right)+q$ :
On the same set of axes, plot the following graphs:
$a\left(\theta \right)=tan\theta -2$
$b\left(\theta \right)=tan\theta -1$
$c\left(\theta \right)=tan\theta $
$d\left(\theta \right)=tan\theta +1$
$e\left(\theta \right)=tan\theta +2$
Use your results to deduce the effect of
$q$ .
On the same set of axes, plot the following graphs:
$f\left(\theta \right)=-2\xb7tan\theta $
$g\left(\theta \right)=-1\xb7tan\theta $
$h\left(\theta \right)=0\xb7tan\theta $
$j\left(\theta \right)=1\xb7tan\theta $
$k\left(\theta \right)=2\xb7tan\theta $
Use your results to deduce the effect of
$a$ .
You should have found that the value of
$a$ affects the steepness of each of the branches. The larger the absolute magnitude of
a , the quicker the branches approach their asymptotes, the values where they are not defined. Negative
$\mathit{a}$ values switch the direction of the branches.
You should have also found that the value of
$q$ affects the vertical shift as for
$sin\theta $ and
$cos\theta $ .
These different properties are summarised in
[link] .
Table summarising general shapes and positions of graphs of functions of the form
$y=atan\left(x\right)+q$ .
$a>0$
$a<0$
$q>0$
$q<0$
Domain and range
The domain of
$f\left(\theta \right)=atan\left(\theta \right)+q$ is all the values of
$\theta $ such that
$cos\theta $ is not equal to 0. We have already seen that when
$cos\theta =0$ ,
$tan\theta =\frac{sin\theta}{cos\theta}$ is undefined, as we have division by zero. We know that
$cos\theta =0$ for all
$\theta ={90}^{\circ}+{180}^{\circ}n$ , where
$n$ is an integer. So the domain of
$f\left(\theta \right)=atan\left(\theta \right)+q$ is all values of
$\theta $ , except the values
$\theta ={90}^{\circ}+{180}^{\circ}n$ .
The range of
$f\left(\theta \right)=atan\theta +q$ is
$\left\{f\right(\theta ):f(\theta )\in (-\infty ,\infty \left)\right\}$ .
Intercepts
The
$y$ -intercept,
${y}_{int}$ , of
$f\left(\theta \right)=atan\left(x\right)+q$ is again simply the value of
$f\left(\theta \right)$ at
$\theta ={0}^{\circ}$ .
As
$\theta $ approaches
${90}^{\circ}$ ,
$tan\theta $ approaches infinity. But as
$\theta $ is undefined at
${90}^{\circ}$ ,
$\theta $ can only approach
${90}^{\circ}$ , but never equal it. Thus the
$tan\theta $ curve gets closer and closer to the line
$\theta ={90}^{\circ}$ , without ever touching it. Thus the line
$\theta ={90}^{\circ}$ is an asymptote of
$tan\theta $ .
$tan\theta $ also has asymptotes at
$\theta ={90}^{\circ}+{180}^{\circ}n$ , where
$n$ is an integer.
Graphs of trigonometric functions
Using your knowldge of the effects of
$a$ and
$q$ , sketch each of the following graphs, without using a table of values, for
$\theta \in [{0}^{\circ};{360}^{\circ}]$
$y=2sin\theta $
$y=-4cos\theta $
$y=-2cos\theta +1$
$y=sin\theta -3$
$y=tan\theta -2$
$y=2cos\theta -1$
Give the equations of each of the following graphs:
The following presentation summarises what you have learnt in this chapter.
Summary
We can define three trigonometric functions for right angled triangles: sine (sin), cosine (cos) and tangent (tan).
Each of these functions have a reciprocal: cosecant (cosec), secant (sec) and cotangent (cot).
We can use the principles of solving equations and the trigonometric functions to help us solve simple trigonometric equations.
We can solve problems in two dimensions that involve right angled triangles.
For some special angles, we can easily find the values of sin, cos and tan.
We can extend the definitions of the trigonometric functions to any angle.
Trigonometry is used to help us solve problems in 2-dimensions, such as finding the height of a building.
We can draw graphs for sin, cos and tan
End of chapter exercises
Calculate the unknown lengths
In the triangle
$PQR$ ,
$PR=20$ cm,
$QR=22$ cm and
$P\widehat{R}Q={30}^{\circ}$ . The perpendicular line from
$P$ to
$QR$ intersects
$QR$ at
$X$ . Calculate
the length
$XR$ ,
the length
$PX$ , and
the angle
$Q\widehat{P}X$
A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder?
A ladder of length 25 m is resting against a wall, the ladder makes an angle
${37}^{\circ}$ to the wall. Find the distance between the wall and the base of the ladder?
In the following triangle find the angle
$A\widehat{B}C$
In the following triangle find the length of side
$CD$
$A(5;0)$ and
$B(11;4)$ . Find the angle between the line through A and B and the x-axis.
$C(0;-13)$ and
$D(-12;14)$ . Find the angle between the line through C and D and the y-axis.
A
$5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ ladder is placed
$2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ from the wall. What is the angle the ladder makes with the wall?
Given the points: E(5;0), F(6;2) and G(8;-2), find angle
$F\widehat{E}G$ .
An isosceles triangle has sides
$9\phantom{\rule{0.166667em}{0ex}}\mathrm{cm},\phantom{\rule{0.166667em}{0ex}}9\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ and
$2\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ . Find the size of the smallest angle of the triangle.
A right-angled triangle has hypotenuse
$13\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ . Find the length of the other two sides if one of the angles of the triangle is
${50}^{\circ}$ .
One of the angles of a rhombus (
rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter
$20\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ is
${30}^{\circ}$ .
Find the sides of the rhombus.
Find the length of both diagonals.
Captain Hook was sailing towards a lighthouse with a height of
$10\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ .
If the top of the lighthouse is
$30\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ away, what is the angle of elevation of the boat to the nearest integer?
If the boat moves another
$7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
(Tricky) A triangle with angles
${40}^{\circ},\phantom{\rule{0.166667em}{0ex}}{40}^{\circ}$ and
${100}^{\circ}$ has a perimeter of
$20\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ . Find the length of each side of the triangle.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?