1. \(\sin 30˚ = \dfrac{1}{2}\) ; \(\sin 30˚ = \dfrac{\dfrac{1}{\sqrt 3}}{\dfrac{2}{\sqrt 3}}\) ;
\(\sin 30˚ = \dfrac{1}{2}\)
Note that the ratios are the same: \(\dfrac{1}{2}\)
\(\cos 30˚ = \dfrac{\sqrt 3}{2}\); \(\cos 30˚ = \dfrac{1}{\dfrac{2}{\sqrt 3}}\);
\(\cos 30˚ = \dfrac{\dfrac{\sqrt 3}{2}}{1}\)
Note that the ratios are the same: \(\dfrac{\sqrt 3}{2}\)
2. \(\sin 60˚ = \dfrac{\sqrt 3}{2}\) ;\(\sin 60˚= \dfrac{1}{\dfrac{2}{\sqrt 3}}\); \(\sin 60˚ = \dfrac{\sqrt 3}{2}\). Note: The ratios are all \(\dfrac{\sqrt 3}{2}\).
\(\cos 60˚ = \dfrac{1}{2}\); \(\cos 60˚ = \dfrac{\dfrac{1}{\sqrt 3}}{\dfrac{2}{\sqrt 3}}\);
\(\cos 60˚ = \dfrac{1}{2}\).
Note that the ratios are the same: \(\dfrac{1}{2}\)
3. \(\sin 45˚ = \dfrac{1}{\sqrt 2}\); \(\cos 45˚ = \dfrac{1}{\sqrt 2}\) or \(\dfrac{\sqrt 2}{2}\)
\(\sin 45˚ =\dfrac{1}{\sqrt 2}\); \(\cos 45˚=\dfrac{1}{\sqrt2}\) or \(\dfrac{\sqrt 2}{2}\)
4. The legs of the right triangle are the same length.
5.
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| Θ | ||||||
| 30° | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt3}{2}\) | \(\dfrac{\sqrt3}{3}\) | 2 | \(\dfrac{2}{√3}\) | \(\sqrt3\) |
| 60° | \(\dfrac{\sqrt3}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt3\) | \(\dfrac{2\sqrt3}{3}\) | 2 | \(\dfrac{\sqrt3}{3}\) |
| 45° | \(\dfrac{\sqrt2}{2}\) | \(\dfrac{\sqrt2}{2}\) | 1 | \(\sqrt2\) | \(\sqrt2\) | 1 |
6. Note that \(\sin 30˚ = \cos 60˚\); \(\sin 60˚ = \cos 30˚\); \(\tan 30˚ = \dfrac1{\tan 60˚}\);
\(\sin 45˚ = \cos45˚\) ;
Reciprocals: \(\csc Θ = \dfrac{1}{\sin Θ}\); \(\sec Θ =\dfrac{1}{\cos Θ}\);
\(\cot Θ = \dfrac{1}{\tan Θ}\)
7. You probably learned SOHCAHTOA in high school.
Return to Practice (Ratios for Special Right Triangles)