Trigonometry / Unit 2: Right Triangle Trigonometry
2.2 Special Right Triangles
Notes
Two triangles give exact trig values.
- 45-45-90: legs $x$, $x$ and hypotenuse $\displaystyle x\sqrt{2}$.
- 30-60-90: short leg $x$ (opposite $30^\circ$), long leg $\displaystyle x\sqrt{3}$ (opposite $60^\circ$), hypotenuse $2x$.
- From these: $\displaystyle \sin 30^\circ = \frac{1}{2}$, $\displaystyle \cos 30^\circ = \frac{\sqrt{3}}{2}$, $\displaystyle \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$, $\displaystyle \sin 60^\circ = \frac{\sqrt{3}}{2}$, $\displaystyle \tan 60^\circ = \sqrt{3}$.
- Example: if the hypotenuse of a 30-60-90 triangle is 8, the short leg is 4 and the long leg is $\displaystyle 4\sqrt{3}$.
Practice
What is the exact value of $\sin 60^\circ$?
In a 30-60-90 triangle, the side opposite $60^\circ$ is $\displaystyle x\sqrt{3}$ and the hypotenuse is $2x$.
A 45-45-90 triangle has legs of length 7. What is the hypotenuse?
Hypotenuse $\displaystyle = \text{leg} \cdot \sqrt{2}$.
A 30-60-90 triangle has hypotenuse 10. What is the length of the long leg?
The short leg is $\displaystyle \frac{10}{2} = 5$ and the long leg is $\displaystyle 5\sqrt{3}$.
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