WORKED EXAMPLE

CIRCULAR

MEASURES

#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.


Explanations
1
2
3
  Download onepager (.pdf)
  Add to favorites See all explanations

SHARE ONEPAGER

 

PRACTICE
[TASKS ACCORDING TO THIS SUBTOPIC]

#Introduction to trigonometry

TASK

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

WANT TO TRY SOME TASKS?
TRY THEM FOR FREE

Sign up for free
Other Onepagers in #Introduction to trigonometry
All explanations on this onepager
#1/ 3

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° is equal to 2π rad, and the half of the circle, 180° is equal to π rad. We can express any angles given in degrees using radians. For example 90° is equal to π2  rad

 

We can convert between these two types of measurements. π rad= 180° 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 π and multiply by 180.

 

 degree=180×radianπ

 

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

 

radian=π×degree180

 

You don’t need to memorize these rules above, because if you know that 180° is equal to π rad you will be able to deduce them.

Example #1

Convert 54° to radian!

1st step: identify what are we looking for?

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

radian=π×degree180

 

2nd step: substitute into the formula:

radian=π×54180=170180=0.94 rad 

Example #2

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=180×radianπ


2nd step: substitute into the formula:

degree=180×2.3π=414π=132° 

#2/ 3

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 π2 radians.

If an angle is greater than 360 degrees / 2π  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°+180°=540° or 2π rad+π rad= 3π 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 sin(x) and the projection of the endpoint of the hand belonging to the same angle x on the horizontal axis is cos(x).

 

In other words cos(x) is the x coordinate of the end point of the hand and sin(x) is the y coordinate of the end point of the hand.

Example #1

Solved the equation:  2cos(x)=1 on the interval x(-180°; 180°)!


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

2cos(x)=1cos(x)=12

 

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

The amount of x=12 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: cos(60°)=12   and to find the other angle, you only need to subtract the 60° form the 360°360°-60°=300°. This part you can figure out by taking a good look at the unit circle.

 

4th step: the final answer is x1=60°, x2=300°

#3/ 3

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, 12, 12, 32  and 1.

 

They are always in pairs, for example

when sin(30°)=12  then cos(30°)=32  

when cos(60°)=12  then sin(60°)=32.

For 45° they both give the same value, that is 12. The other two options, 0 and 1 are easy to read from the unit circle.

 

The values for tan(x) can be calculated by using the values of sine and cosine and considering tan(x) as sin(x)cos(x).

Example #1

Use the unit circle to find tan(225°)!


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

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

 

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

From the first look, 225° 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 tan(45°)=1 so tan(225°)=1.

 

Example #2

Don't use the unit circle to find tan(225°)


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: 

tan(x)=sin(x)cos(x)

 

sin(45°) and cos(45°) are both equal to 12 , but we are in the 3rd quadrant which means that both of the values has a negative sign : sin(225°)=-12 , cos(225°)=-12  

 

let’s put these together: 

tan(225°)sin(225°)cos(225°)=-12-12=1