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. Ignore the last slide.
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.
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Rafiq
Rafiq
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Rafiq
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