Trigonometric functions

Recall the three trigonometric relationships in right-angled triangles (SOHCAHTOA):

See the figure below (left). Here, $H$, $O$ and $A$ denote the hypotenuse, opposite side and adjacent side. Note that $0^\circ<\alpha<90^\circ$ for the relationships to make sense. At the moment it is not clear what $\sin(140^\circ)$ means, as there is no right angled triangle with such an angle. But ultimately we want to be able to calculate $\sin(\alpha), \cos(\alpha)$ and $\tan(\alpha)$ for $\alpha=140^\circ$, or for any other angle of any size. This will lead to the trigonometric functions.

First, let's start by considering again right-angled triangles, but this time with hypotenuse $H=1$ (figure above, middle). We then have

and

Thus, if $H=1$, the sine is the length of the vertical side, and the cosine is length of the horizontal side. By the way, we can also represent $\tan(\alpha)$ as a side length. To do so, we have to extend the triangle so that the horizontal side has length $1$ (figure above, right). The vertical side $u$ of the new triangle is then $\tan(\alpha)$, because

Extending the definition of the sine and cosine

To extend the $\sin(\alpha)$ and $\cos(\alpha)$ to arbitrary values of $\alpha$, we focus on the fact we have observed above, namely that for $H=1$ the $\sin(\alpha)$ is the vertical side length and the $\cos(\alpha)$ is the horizontal side length. So let us introduce a new definition of the sine and cosine. It involves the unit circle.

1. Draw a unit circle, and indicate a vertical and a horizontal axis, both passing through the centre of the circle (see figure below). Each arbitrary angle $\alpha$ defines the position of a point $P$ on the unit circle. From now on we use the arc length $x$ rather than the angle $\alpha$ to describe the position of this point $P$.

2. For a given value of $x$ (the input to the machine) determine the corresponding point $P$ on the unit circle, and define the following output $y$:

   $$\boxed{\begin{array}{lll} y=\sin(x)&=&\text{distance between $P$ and horizontal axis ("rise")}\\ y=\cos(x)&=&\text{distance between $P$ and the vertical axis ("run")} \end{array}}$$

   In addition, "rise" is negative if $P$ is below the horizontal axis, and "run" is negative if $P$ is on the left side of the vertical axis (figure below, right).

It should be clear that for $x$ between $0$ and $\frac{\pi}{2}$, or for $\alpha$ between $0^\circ$ and $90^\circ$, these definitions are simply the original trigonometric relationships SOHCAHTOA where $H=1$:

The function $\tan(x)$ can be defined by the length of the vertical line of the extended triangles (see top of section). Or we can simply set

As a summary, we represent the three trigonometric functions $\sin(x), \cos(x)$ and $\tan(x)$ as machines. The input $x$ is interpreted as the arc length (radians):

Exercise 1

1. No calculator. Determine $\sin(x)$, $\cos(x)$, and $\tan(x)$ for the following $x$-values: $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$, $\frac{5\pi}{2}$, $-\frac{\pi}{2}$.

2. Without using the trigonometric function keys sin, cos, and tan on the calculator, determine $\sin(x)$, $\cos(x)$, and $\tan(x)$ for the following $x$-values: $\frac{\pi}{4}$, $\frac{\pi}{6}$, $\frac{7\pi}{4}$, $\frac{-5\pi}{6}$. Verify your results using the calculator.

3. Without using trigonometric function keys sin^-1 and cos^-1 on the calculator, give an estimate of $x$ such that $\sin(x)=0.75$. Also, if $\sin(x)=0.75$, what is the exact value of $\sin(x+\pi)$?

4. No calculator: sketch the graph of the three trigonometric functions. Use the unit circle and indicate the points on the graph at $x=0, \frac{\pi}{2},\frac{3\pi}{2}$ and $2\pi$. Check the graphs using Geogebra. An animation illustrating the relationship between the unit circle and the graph of the sine and cosine functions can be found here: animation. Note: recall that $\frac{0}{1}=\frac{0}{-1}=0$ and $\frac{1}{0}=\infty$ and $\frac{-1}{0}=-\infty$.

5. No calculator. Express the functions $f(x)=\sin(x+\frac{\pi}{2})$ and $g(x)=\sin(x-\frac{\pi}{2})$ using $\cos(x)$.

6. No calculator. What is $\sin^2(x)+\cos^2(x)$ for every $x$? Note: the notation $\sin^2(x)$ is a short form for $\sin(x)\cdot \sin(x)=(\sin(x))^2$.

7. With calculator. Find an $x$-intercept of the functions $f$ and $g$ by solving the equations $f(x)=0$ and $g(x)=0$.

   1. $f(x)=2\cos(2x+1)+1$
   2. $g(x)=0.5 \tan(3x)-10$

8. With calculator. Find an intersection point between the functions $f$ and $g$

   1. $f(x)=\sin(x)$, $g(x)=0.4$
   2. $f(x)=3\cos(2x-0.5)$, $g(x)=0.1$
   3. $f(x)=\sin(3x)$, $g(x)=2\cdot \cos(3x)$

9. Given $\sin(x)=0.6$, we need the arc sine to find $x$:

   $$x=\arcsin(0.6)=\sin^{-1}(0.6)$$

   Why is there the name "arc" in the name? Argue with the unit circle.

10. Without calculator. Find the solutions of

    1. $\cos(2x)=-1$

    2. $2\sin(3x-1)=0$

Solution

10

1. Find $x$ with $2x=\pi$, or $2x=5\pi$, and so on, so $x=\frac{\pi}{2}$, or $x=\frac{5\pi}{2}$, ...
2. Find $x$ with $3x-1=0$ or $3x-1=\pi$, ..., so $x=\frac{1}{3}$, or $x=\frac{\pi+1}{3}$, ... .