#Introduction to trigonometry

Trigonometry is a slightly longer topic of IB Math than the usual, but starting from the basics, you can build your knowledge. This onepager introduces sine, cosine and tangent.

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#Introduction to trigonometry

Write down the following expressions in terms of cos(x).

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**Radian**

You already know that we are measuring angles in degrees, but in mathematics there is another type of measurement for angles and that is called radian.

One whole circle, $360\xb0$ is equal to $2\pi rad$, and the half of the circle, $180\xb0$ is equal to $\mathrm{\pi}\mathrm{rad}$. We can express any angles given in degrees using radians. For example $90\xb0$ is equal to $\frac{\mathrm{\pi}}{2}rad$.

We can convert between these two types of measurements. $\mathrm{\pi}\mathrm{rad}=180\xb0$ is the conversion number, similarly to converting hours to minutes with the conversion number $1$ hour$=60$ minutes.

If we want to change radian to degree, first we have to divide by $\pi $ and multiply by $180$.

$degree=\frac{180\times radian}{\mathrm{\pi}}$

If we want to change degree to radian, first we have to divide by $180$ and multiply with $\pi $.

$radian=\frac{\mathrm{\pi}\times degree}{180}$

You don’t need to memorize these rules above, because if you know that $180\xb0$ is equal to $\mathrm{\pi}\mathrm{rad}$ you will be able to deduce them.

**Convert 54° to radian!**

**1st step**: identify what are we looking for?

We are looking for the **radian**. Select the correct expression:

$radian=\frac{\mathrm{\pi}\times degree}{180}$

**2nd step**: substitute into the formula:

$radian=\frac{\mathrm{\pi}\times 54}{180}=\frac{170}{180}=0.94rad$

**Convert 2.3 rad to degree!**

**1st step**: identify what are we looking for?

We are looking for the **degree**. Select the correct expression:

$degree=\frac{180\times radian}{\mathrm{\pi}}$

**2nd step**: substitute into the formula:

$degree=\frac{180\times 2.3}{\mathrm{\pi}}=\frac{414}{\mathrm{\pi}}=132\xb0$

**Unit circle**

Let’s see a circle with center $O$ and radius $1$. This is the unit circle. With the unit circle we can measure any angles.

There's a hand going around anti-clockwise. The measurement starts from the positive side of the horizontal axis, so that is $0$ degrees. When it is on the positive side of the vertical axis, it becomes $90$ degrees, or $\frac{\mathrm{\pi}}{2}$ radians.

If an angle is greater than $360$ degrees / $2\mathrm{\pi}$ radians, then it just means that the hand went around and it starts from $0$ again. If we take $1$ and a half circle we measured $360\xb0+180\xb0=540\xb0$ or $2\mathrm{\pi}\mathrm{rad}+\mathrm{\pi}\mathrm{rad}=3\mathrm{\pi}\mathrm{rad}$.

Using the unit circle you can find the sine and cosine of an angle. The projection of the end point of the hand belonging to an angel $x$ on the vertical axis is $\mathrm{sin}\left(x\right)$ and the projection of the endpoint of the hand belonging to the same angle $x$ on the horizontal axis is $\mathrm{cos}\left(x\right)$.

In other words $\mathrm{cos}\left(x\right)$ is the $x$ coordinate of the end point of the hand and $\mathrm{sin}\left(x\right)$ is the $y$ coordinate of the end point of the hand.

**Solved the equation: $2\mathrm{cos}\left(x\right)=1$ on the interval $x\in (-180\xb0;180\xb0)$!**

**1st step**: rearrange the equation to get a single trigonometric function on one side and a constant (just a number) on the other side

$2\mathrm{cos}\left(x\right)=1\phantom{\rule{0ex}{0ex}}\mathrm{cos}\left(x\right)=\frac{1}{2}$

**2nd step**: draw the unit circle as accurate as you can to be able to read the values better.

The amount of $x=\frac{1}{2}$ belongs to $2$ different angles, one in the first quadrant and one in the fourth quadrant.

**3rd step**: now you have to remember the correct angle from the "known values" table: $\mathrm{cos}(60\xb0)=\frac{1}{2}$ and to find the other angle, you only need to subtract the $60\xb0$ form the $360\xb0$, $360\xb0-60\xb0=300\xb0$. This part you can figure out by taking a good look at the unit circle.

**4th step**: the final answer is ${x}_{1}=60\xb0,{x}_{2}=300\xb0$

**Exact values**

It can be difficult to solve a trigonometric equation if you are not allowed to use your calculator. This is why we recommend you to memorize these exact values of trigonometric functions.

To be able to memorise the values better, we suggest that you memorise the options for sine and cosine. These are $0,\frac{1}{2},\frac{1}{\sqrt{2}},\frac{\sqrt{3}}{2}$ and $1$.

They are always in pairs, for example

when $\mathrm{sin}(30\xb0)=\frac{1}{2}$ then $\mathrm{cos}(30\xb0)=\frac{\sqrt{3}}{2}$

when $\mathrm{cos}(60\xb0)=\frac{1}{2}$ then $\mathrm{sin}(60\xb0)=\frac{\sqrt{3}}{2}$.

For $45\xb0$ they both give the same value, that is $\frac{1}{\sqrt{2}}$. The other two options, $0$ and $1$ are easy to read from the unit circle.

The values for $\mathrm{tan}\left(x\right)$ can be calculated by using the values of sine and cosine and considering $\mathrm{tan}\left(x\right)$ as $\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$.

**Use the unit circle to find $\mathrm{tan}(225\xb0)$!**

**1st step**: found the quadrant where our angle will be!

It's more than $180\xb0$ but less than $270\xb0$ so it will be in the 3rd quadrant.

**2nd step**: decide which angle you are looking for.

From the first look, $225\xb0$ may not look like any of the memorized values, but if you subtract $180$, then you get $45$.

**3rd step**: draw the unit circle as accurate as you can to be able to read the values better

**4th step**: evaluate!

If you memorized the tangent values you know that $\mathrm{tan}(45\xb0)=1$ so $\mathrm{tan}(225\xb0)=1$.

**Don't use the unit circle to find $\mathrm{tan}(225\xb0)$! **

If you memorized only the sine and cosine values, no problem, because you are able to tell the solution from the definition of tangent function:

$\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$.

$\mathrm{sin}(45\xb0)$ and $\mathrm{cos}(45\xb0)$ are both equal to $\frac{1}{\sqrt{2}}$ , but we are in the 3rd quadrant which means that both of the values has a negative sign : $\mathrm{sin}(225\xb0)=-\frac{1}{\sqrt{2}},\mathrm{cos}(225\xb0)=-\frac{1}{\sqrt{2}}$

let’s put these together:

$\mathrm{tan}(225\xb0)\frac{\mathrm{sin}(225\xb0)}{\mathrm{cos}(225\xb0)}=\frac{-{\displaystyle \frac{1}{\sqrt{2}}}}{-\frac{1}{\sqrt{2}}}=1$