Questions: Use a half-angle formula to find the exact value of the following expression. tan 157.5° tan 157.5°=-0.4142 (Simplify your answer, including any radicals. Use integers or fractions for an)

Solution Steps

To find the exact value of \(\tan 157.5^\circ\), we can use the half-angle identity for tangent. The angle \(157.5^\circ\) is half of \(315^\circ\), so we can use the identity:

\[ \tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} \]

where \(\theta = 315^\circ\). We first need to find \(\cos 315^\circ\) and \(\sin 315^\circ\) using the unit circle, and then apply the half-angle formula.

To find \(\tan 157.5^\circ\), we use the half-angle formula for tangent: \[ \tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} \] where \(\theta = 315^\circ\).

Using the unit circle, we find: \[ \cos 315^\circ = \frac{\sqrt{2}}{2} \quad \text{and} \quad \sin 315^\circ = -\frac{\sqrt{2}}{2} \]

Substituting the values into the half-angle formula: \[ \tan 157.5^\circ = \frac{1 - \cos 315^\circ}{\sin 315^\circ} = \frac{1 - \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} \]

Simplifying the expression: \[ \tan 157.5^\circ = \frac{1 - \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = \frac{\frac{2 - \sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = \frac{2 - \sqrt{2}}{-\sqrt{2}} = -\frac{2 - \sqrt{2}}{\sqrt{2}} = -\frac{2}{\sqrt{2}} + 1 = -\sqrt{2} + 1 \]

Final Answer