Basic Technical Mathematics, 11th Edition

4.3 Values of the Trigonometric Functions

• Function Values Using Geometry • Function Values from Calculator • Inverse Trigonometric Functions • Accuracy of Trigonometric Functions • Reciprocal Functions

We often need values of the trigonometric functions for angles measured in degrees. One way to find some of these values for particular angles is to use certain basic facts from geometry. This is illustrated in the next two examples.

EXAMPLE 1 Function values of 30 °  and 60 ° 

From geometry, we find that in a right triangle, the side opposite a $30\hspace{0.17em}°\hspace{0.17em}$ angle is one-half of the hypotenuse. Therefore, in Fig. 4.18, letting $y\hspace{0.17em}=\hspace{0.17em}1$ and $r\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}$ and using the Pythagorean theorem, $x\hspace{0.17em}=\hspace{0.17em}\sqrt{2^2\hspace{0.17em}-\hspace{0.17em}{1}^2}\hspace{0.17em}=\hspace{0.17em}\sqrt{3}\hspace{0.17em}.\hspace{0.17em}$ Now with $x\hspace{0.17em}=\hspace{0.17em}\sqrt{3}\hspace{0.17em},\hspace{0.17em}y\hspace{0.17em}=\hspace{0.17em}1\hspace{0.17em},\hspace{0.17em}$ and $r\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}$

$$\sin 30\hspace{0.17em}°\hspace{0.17em}\; ...$$