Right Angle Trig

The new syllabus - first exams 2014 - seems to be focusing less on right angle trig. Not sure if students are supposed to be already familiar with the topic or if the IB is trying to make math less relevant and more abstract. Hooray! Either way I see it as vital to understanding more complex trig ideas. I can't promise that these topics will show up on exams. I can promise if you don't understand these ideas trig functions will just be that much harder.

### Similar Triangles

Similar shapes are essentially shapes that are scaled images of each other. For example the triangles below are similar in that the larger triangle has sides that are exactly twice as long as the sides of the smaller triangle. Also the angles of the two triangles are identical!

Now before you blow by this idea too quickly… If a triangle has two perpendicular sides with the ratio of $\frac{2}{6}=\frac{1}{3}$ then the angle opposite the shorter side must be $33.69^\circ$ (allowing for rounding error).

We can make statements like this all day long. In the end we can conclude that knowing the angle (of a right triangle) tells us something about the ratio of two sides and vice versa.

That was 2-3 weeks of geometry. I apologize for going so fast.

### Definitions

Talking about triangles with their sides and angles gets a bit tough without names and such… So lets try to clarify. We define three sides based on their relative position to the angle we are "interested" in.

From this we define three basic ratios. There are actually six but the other 3 are simply reciprocals and not often talked about in Math SL.

(1)
\begin{align} sin \theta = \frac {opposite}{hypotenuse} \end{align}
(2)
\begin{align} cos \theta = \frac {adjacent}{hypotenuse} \end{align}
(3)
\begin{align} tan \theta = \frac {opposite}{adjacent} \end{align}

Note that $sin \theta$ is really function notation. This is NOT "sin" multiplied by $\theta$. When things get more complex the three trig functions are written with parentheses which makes the function properties a bit more obvious. For example $sin(\theta)$.