Right-Angled Trigonometry

Starting point: Years 9 and 10

Pythagoras' theorem

In a right-angled triangle with hypotenuse $c$ (the longest side, opposite the right angle):

$$a^2+b^2=c^2$$

So a triangle with sides 6 and 8 has hypotenuse $\sqrt{36+64}=10$.

Naming the sides

Pick an angle $\theta$. Then:

• Hypotenuse (H): the longest side, opposite the right angle.
• Opposite (O): across from $\theta$.
• Adjacent (A): next to $\theta$, not the hypotenuse.

The three ratios

Angles of elevation and depression

The angle of elevation is measured up from the horizontal. The angle of depression is measured down from the horizontal. They are equal for two people looking at each other, because they are alternate angles between parallel lines.

Bearings

A true bearing is a three-digit angle measured clockwise from north. East is $090^{\circ }$, south is $180^{\circ }$, west is $270^{\circ }$. A compass bearing like S$40^{\circ }$W means start facing south and turn $40^{\circ }$ toward west.

Quick summary

• Pythagoras: $a^2+b^2=c^2$, with $c$ the hypotenuse.
• SOH CAH TOA links an angle to two sides.
• Use inverse trig (${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$) to find angles.
• Elevation is up from the horizontal, depression is down. True bearings go clockwise from north.
• Exact values come from the $45^{\circ }$ and $30^{\circ }$-$60^{\circ }$ triangles.

Quiz