Functions of Special Angles
Evaluate trigonometric functions for 30-60-90 and 45-45-90 special right triangles using side length ratios and exact values in Precalculus.
Key Concepts
Property When dealing with right triangles, we can evaluate trigonometric functions of special angles using side length ratios. A $30^\circ, 60^\circ, 90^\circ$ triangle, also known as a $\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}$ triangle, has side lengths in the relation $s, s\sqrt{3}, 2s$. A $45^\circ, 45^\circ, 90^\circ$ triangle, also known as a $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ triangle, has side lengths in the relation $s, s, s\sqrt{2}$. We can use these ratios to find exact values.
Examples To find the value of $\sin(45^\circ)$, we use the side ratios of a $45^\circ 45^\circ 90^\circ$ triangle ($s, s, s\sqrt{2}$). Thus, $\sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{s}{s\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.
To evaluate $\cos(60^\circ)$, we use a $30^\circ 60^\circ 90^\circ$ triangle. The side adjacent to the $60^\circ$ angle is $s$ and the hypotenuse is $2s$. So, $\cos(60^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{s}{2s} = \frac{1}{2}$.
Common Questions
What are the special right triangles in trigonometry?
The two special right triangles are the 30-60-90 triangle with side ratios s : s√3 : 2s, and the 45-45-90 triangle with side ratios s : s : s√2. These triangles produce exact trigonometric values without needing a calculator.
What are the exact trig values for 30°, 45°, and 60°?
For 30°: sin = 1/2, cos = √3/2, tan = 1/√3. For 45°: sin = cos = √2/2, tan = 1. For 60°: sin = √3/2, cos = 1/2, tan = √3. Memorizing these values is essential for precalculus and calculus.
Why are 30-60-90 and 45-45-90 triangles called special?
These triangles are called special because their angle measures always produce exact, predictable side length ratios using radicals. This allows you to find exact trigonometric function values rather than decimal approximations, which is critical in advanced math and physics applications.