Question

If \(630^\circ<\theta<810^\circ\) and \(\tan \theta = -\frac{7}{24}\), then find \(\cos \left(\frac{\theta}{4}\right)\).

• A. \(-\sqrt{\frac{7 + 5\sqrt{2}}{10 \sqrt{2}}}\)
• B. \(\sqrt{\frac{7 + 5\sqrt{2}}{2 \sqrt{2}}}\)
• C. \(-\sqrt{\frac{5\sqrt{2} - 7}{10 \sqrt{2}}}\)
• D. \(\sqrt{\frac{5\sqrt{2} - 7}{2 \sqrt{2}}}\)

Hint:

Use quadrant analysis and half-angle formulas for trigonometric simplification.

Answer 1

The Correct Option is A

Step 1: Find quadrant of \(\theta\)
Given \(630^\circ<\theta<810^\circ\), which lies in the fourth quadrant (since \(630^\circ = 270^\circ + 360^\circ\)). Step 2: Use \(\tan \theta = -\frac{7}{24}\)
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{7}{24} \] Step 3: Find \(\cos \theta\)
Using Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \left(\frac{-7}{25}\right)^2 + \cos^2 \theta = 1 \] (since \(\tan \theta = \frac{-7}{24}\), the hypotenuse is 25) Calculate: \[ \cos \theta = \pm \frac{24}{25} \] In the fourth quadrant, cosine is positive, so: \[ \cos \theta = \frac{24}{25} \] Step 4: Use half-angle formula for \(\cos \frac{\theta}{4}\)
\[ \cos \frac{\theta}{4} = \pm \sqrt{\frac{1 + \cos \frac{\theta}{2}}{2}} \] Use multiple half-angle reductions to find exact value, eventually simplifying to: \[ -\sqrt{\frac{7 + 5\sqrt{2}}{10 \sqrt{2}}} \]