Special Triangles

Section 8.3 Special Triangles

Certain nice angles appear in triangles with pleasing geometry which allows us to compute their trigonometric ratios qexactly.

Example 8.21. The 45-45-90 Triangle.

Start with a square. You may label the side lengths any value you want, but \(1\) is convenient. All the angles are \(\pi/2 = 90^\circ\text{.}\)

Draw a diagonal with length \(\sqrt{1^2+1^2} = \sqrt{2}\text{.}\) The new angles are \(\pi/4 = 45^\circ\text{.}\)

This forms a “45-45-90” triangle. From this, we can compute exactly any trigonometric function value for \(\pi/4\) (radians) or equivalently \(45^\circ\text{.}\)

\begin{align*} &\sin(\pi/4) = \sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\cos(\pi/4) = \cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\tan(\pi/4) = \tan(45^\circ) = \frac{1}{1} = 1\\ &\csc(\pi/4) = \csc(45^\circ) = \frac{\sqrt{2}}{1} = \sqrt{2}\\ &\sec(\pi/4) = \sec(45^\circ) = \frac{\sqrt{2}}{1} = \sqrt{2}\\ &\cot(\pi/4) = \cot(45^\circ) = \frac{1}{1} = 1 \end{align*}

Example 8.23. The 30-60-90 Triangle.

Start with an equilateral triangle, all sides and all angles are equal to \(\pi/3 = 60^\circ\text{.}\) You can label the side lengths whatever you like, but its convenient to start with length \(2\text{.}\)

Divide the triangle into two right triangles by bisecting one of the angles into two \(\pi/6 = 30^\circ\) angles. This divides the opposite side into two equal parts of length \(1\text{.}\)

Finally, the missing height can be found with the Pythagorean Theorem

\begin{equation*} (\text{height})^2 + 1^2 = 2^2 \end{equation*}

so that

\begin{equation*} (\text{height}) = \sqrt{4-1}=\sqrt{3}. \end{equation*}

This gives two “30-60-90” triangles which allows us to find exact values of trigonometric functions for \(30^\circ = \pi/6\) and its complementary angle \(60^\circ = \pi/3\text{.}\) We’ll summarize in radians:

\begin{align*} &\sin(\pi/6) = \cos(\pi/3) = \frac{1}{2}\\ &\cos(\pi/6) = \sin(\pi/3) = \frac{\sqrt{3}}{2}\\ &\tan(\pi/6) = \cot(\pi/3) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\\ &\csc(\pi/6) = \sec(\pi/3) = 2\\ &\sec(\pi/6) = \csc(\pi/3) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\\ &\cot(\pi/6) = \tan(\pi/3) = \sqrt{3} \end{align*}