Definitions, simple applications, and graphs of trigonometric  (Page 7/7)

Let us look back at our values for $tan\theta$

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

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 $y=atan\left(\theta \right)+q$ :

1. On the same set of axes, plot the following graphs:
   1. $a\left(\theta \right)=tan\theta -2$
   2. $b\left(\theta \right)=tan\theta -1$
   3. $c\left(\theta \right)=tan\theta$
   4. $d\left(\theta \right)=tan\theta +1$
   5. $e\left(\theta \right)=tan\theta +2$
   Use your results to deduce the effect of $q$ .
2. On the same set of axes, plot the following graphs:
   1. $f\left(\theta \right)=-2·tan\theta$
   2. $g\left(\theta \right)=-1·tan\theta$
   3. $h\left(\theta \right)=0·tan\theta$
   4. $j\left(\theta \right)=1·tan\theta$
   5. $k\left(\theta \right)=2·tan\theta$
   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] .

	$a>0$	$a<0$
$q>0$
$q<0$

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 $\theta =0^{\circ }$ .

Asymptotes

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

1. Using your knowldge of the effects of $a$ and $q$ , sketch each of the following graphs, without using a table of values, for $\theta \in [0^{\circ };{360}^{\circ }]$
   1. $y=2sin\theta$
   2. $y=-4cos\theta$
   3. $y=-2cos\theta +1$
   4. $y=sin\theta -3$
   5. $y=tan\theta -2$
   6. $y=2cos\theta -1$
2. 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

1. Calculate the unknown lengths

   

2. 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
   1. the length $XR$ ,
   2. the length $PX$ , and
   3. the angle $Q\widehat{P}X$
3. 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?
   
4. 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?
   
5. In the following triangle find the angle $A\widehat{B}C$

   

6. In the following triangle find the length of side $CD$

   

7. $A(5;0)$ and $B(11;4)$ . Find the angle between the line through A and B and the x-axis.
   
8. $C(0;-13)$ and $D(-12;14)$ . Find the angle between the line through C and D and the y-axis.
   
9. A $5\mathrm{m}$ ladder is placed $2\mathrm{m}$ from the wall. What is the angle the ladder makes with the wall?
   
10. Given the points: E(5;0), F(6;2) and G(8;-2), find angle $F\widehat{E}G$ .
    
11. An isosceles triangle has sides $9\mathrm{cm},9\mathrm{cm}$ and $2\mathrm{cm}$ . Find the size of the smallest angle of the triangle.
    
12. A right-angled triangle has hypotenuse $13\mathrm{mm}$ . Find the length of the other two sides if one of the angles of the triangle is ${50}^{\circ }$ .
    
13. One of the angles of a rhombus ( rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter $20\mathrm{cm}$ is ${30}^{\circ }$ .
    1. Find the sides of the rhombus.
    2. Find the length of both diagonals.
14. Captain Hook was sailing towards a lighthouse with a height of $10\mathrm{m}$ .
    1. If the top of the lighthouse is $30\mathrm{m}$ away, what is the angle of elevation of the boat to the nearest integer?
    2. If the boat moves another $7\mathrm{m}$ towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
15. (Tricky) A triangle with angles ${40}^{\circ },{40}^{\circ }$ and ${100}^{\circ }$ has a perimeter of $20\mathrm{cm}$ . Find the length of each side of the triangle.