The steps to obtain the six trigonometric ratios of the special angles \( 30^{\circ}, 45^{\circ} \) and \( 60^{\circ} \) using special right triangles are presented.
The values of the six trigonometric of the special angles are given in a table at the bottom of the page.
It is a right triangle with equal sides and angles equal to \( 45^{\circ} \) as shown in figure 1 below.
We start with an equilateral triangle with side \( a \) as shown in figure 2 below. Then draw a perpendicular from one of the vertices of the triangle to the opposite base. This perpendicular bisects the angle into two equal angles of \( 30^{\circ} \) and the opposite side into two equal segments of length \( \dfrac{a}{2} \) as shown in figure 3 below.
We now use the same triangle in figure 3 to find all six trigonometric ratios of \( 60^{\circ} \).
\( \qquad \sin 60^{\circ} = \dfrac{\text{Opposite Side}}{\text{Hypotenuse}} = \dfrac{h}{a} = \dfrac{\dfrac{a \sqrt 3}{2}}{a} = \dfrac{\sqrt 3}{2} \)
\( \qquad \cos 60^{\circ} = \dfrac{\text{Adjacent Side}}{\text{Hypotenuse}} = \dfrac{a/2}{a} = \dfrac{1}{2} \)
\( \qquad \tan 60^{\circ} = \dfrac{\text{Opposite Side}}{\text{Adjacent Side}} = \dfrac{h}{a/2} = \dfrac{\dfrac{a \sqrt 3}{2}}{a/2} = \sqrt 3 \)
\( \qquad \csc 60^{\circ} = \dfrac{1}{\sin 60^{\circ}} = \dfrac{1}{\dfrac{\sqrt 3}{2}} = \dfrac{2}{\sqrt 3} \)
\( \qquad \sec 60^{\circ} = \dfrac{1}{\cos 60^{\circ}} = \dfrac{1}{1/2} = 2\)
\( \qquad \cot 60^{\circ} = \dfrac{ 1 }{\tan 60^{\circ} } = \dfrac{1}{\sqrt 3} \)
Here we group all values of the six trigonometric function in a table.
NOTE that the letter \( U \) used in the table mean undefined.
\( \color{red}{\theta \;\;\; \text{( in Degrees )} }\) | \( \color{red}{ 0^{\circ} } \) | \( \color{red}{ 30^{\circ} } \) | \( \color{red}{ 45^{\circ} } \) | \( \color{red}{ 60^{\circ} } \) | \( \color{red}{ 90^{\circ} } \) |
\( \color{red}{\theta \;\;\; \text{( in Radians )} }\) | \( \color{red}{ 0 } \) | \( \color{red}{ \dfrac{\pi}{6} } \) | \( \color{red}{ \dfrac{\pi}{4} } \) | \( \color{red}{ \dfrac{\pi}{3} } \) | \( \color{red}{ \dfrac{\pi}{2} } \) |
\( \color{red}{\sin \theta}\) | \( 0 \) | \( \dfrac{1}{2} \) | \( \dfrac{\sqrt 2}{2} \) | \( \dfrac{\sqrt 3}{2} \) | \( 1 \) |
\( \color{red}{\cos \theta} \) | \( 1 \) | \( \dfrac{\sqrt 3}{2} \) | \( \dfrac{\sqrt 2}{2} \) | \( \dfrac{1}{2} \) | \( 0 \) |
\( \color{red}{ \tan \theta } \) | \( 0 \) | \( \dfrac{1}{\sqrt 3} \) | \( 1 \) | \( \sqrt 3 \) | \( \text{U}\) |
\( \color{red}{\csc \theta}\) | \( \text{U}\) | \( 2 \) | \( \sqrt 2 \) | \( \dfrac{2}{\sqrt 3} \) | \( 1 \) |
\( \color{red}{\sec \theta} \) | \( 1 \) | \( \dfrac{2}{\sqrt 3} \) | \( \sqrt 2 \) | \( 2 \) | \( \text{U}\) |
\( \color{red}{ \cot \theta } \) | \( \text{U}\) | \( \sqrt 3 \) | \( 1 \) | \( \dfrac{1}{\sqrt 3} \) | \( 0 \) |