11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 3/12)

Functions of the form $y=cos(\theta +p)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=cos(\theta -{90}^{\circ })$
2. $b\left(\theta \right)=cos(\theta -{60}^{\circ })$
3. $c\left(\theta \right)=cos\theta$
4. $d\left(\theta \right)=cos(\theta +{90}^{\circ })$
5. $e\left(\theta \right)=cos(\theta +{180}^{\circ })$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ affects the $y$ -intercept and phase shift of the graph. As in the case of the sine graph, positive values of $p$ shift the cosine graph left while negative $p$ values shift the graph right.

These different properties are summarised in [link] .

$p>0$	$p<0$

Domain and range

For $f\left(\theta \right)=cos(\theta +p)$ , the domain is $\{\theta :\theta \in \mathbb{R}\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.

The range of $f\left(\theta \right)=cos(\theta +p)$ is $\left\{f\right(\theta ):f(\theta )\in [-1,1\left]\right\}$ .

Intercepts

For functions of the form, $y=cos(\theta +p)$ , the details of calculating the intercept with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta =0^{\circ }$

Functions of the form $y=tan(\theta +p)$

In the equation, $y=tan(\theta +p)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=tan(\theta +{30}^{\circ })$ .

Functions of the form $y=tan(\theta +p)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=tan(\theta -{90}^{\circ })$
2. $b\left(\theta \right)=tan(\theta -{60}^{\circ })$
3. $c\left(\theta \right)=tan\theta$
4. $d\left(\theta \right)=tan(\theta +{60}^{\circ })$
5. $e\left(\theta \right)=tan(\theta +{180}^{\circ })$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ once again affects the $y$ -intercept and phase shift of the graph. There is a horizontal shift to the left if $p$ is positive and to the right if $p$ is negative.

These different properties are summarised in [link] .

$k>0$	$k<0$

Domain and range

For $f\left(\theta \right)=tan(\theta +p)$ , the domain for one branch is $\{\theta :\theta \in (-{90}^{\circ }-p,{90}^{\circ }-p\}$ because the function is undefined for $\theta =-{90}^{\circ }-p$ and $\theta ={90}^{\circ }-p$ .

The range of $f\left(\theta \right)=tan(\theta +p)$ is $\left\{f\right(\theta ):f(\theta )\in (-\infty ,\infty \left)\right\}$ .

Intercepts

For functions of the form, $y=tan(\theta +p)$ , the details of calculating the intercepts with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta =0^{\circ }$

Asymptotes

The graph of $tan(\theta +p)$ has asymptotes because as $\theta +p$ approaches ${90}^{\circ }$ , $tan(\theta +p)$ approaches infinity. Thus, there is no defined value of the function at the asymptote values.

Functions of various form

Using your knowledge of the effects of $p$ and $k$ draw a rough sketch of the following graphs without a table of values.

1. $y=sin3x$
2. $y=-cos2x$
3. $y=tan\frac{1}{2}x$
4. $y=sin(x-{45}^{\circ })$
5. $y=cos(x+{45}^{\circ })$
6. $y=tan(x-{45}^{\circ })$
7. $y=2sin2x$
8. $y=sin(x+{30}^{\circ })+1$

Trigonometric identities

Deriving values of trigonometric functions for ${30}^{\circ }$ , ${45}^{\circ }$ And ${60}^{\circ }$

Keeping in mind that trigonometric functions apply only to right-angled triangles, we can derive values of trigonometric functions for ${30}^{\circ }$ , ${45}^{\circ }$ and ${60}^{\circ }$ . We shall start with ${45}^{\circ }$ as this is the easiest.

Take any right-angled triangle with one angle ${45}^{\circ }$ . Then, because one angle is ${90}^{\circ }$ , the third angle is also ${45}^{\circ }$ . So we have an isosceles right-angled triangle as shown in [link] .