Let [tex]\frac{3 \pi}{2} \ \textless \ t \ \textless \ 2 \pi[/tex] and [tex]\cos(t) = \frac{3}{4}[/tex]. Use a half-angle formula to find [tex]\cos\left(\frac{t}{2}\right)[/tex]. (Give an exact answer. Use symbolic notation and fractions where needed.)

[tex]\cos\left(\frac{t}{2}\right) =[/tex]

[tex]\square[/tex]



Answer :

To find \(\cos \left( \frac{t}{2} \right)\) given that \(\frac{3\pi}{2} < t < 2\pi\) and \(\cos(t) = \frac{3}{4}\), we can use the half-angle formula for cosine.

The half-angle formula for cosine states:
[tex]\[ \cos \left( \frac{t}{2} \right) = \pm \sqrt{\frac{1 + \cos(t)}{2}} \][/tex]

We need to determine the correct sign for \(\cos \left( \frac{t}{2} \right)\). Since \( \frac{3\pi}{2} < t < 2\pi \), dividing these bounds by 2 gives \( \frac{3\pi}{4} < \frac{t}{2} < \pi \).

The cosine function is negative in the interval \( \left( \frac{3\pi}{4}, \pi \right) \). Therefore, \(\cos \left( \frac{t}{2} \right) \) will be negative in this range.

Now, substitute \(\cos(t) = \frac{3}{4}\) into the half-angle formula:
[tex]\[ \cos \left( \frac{t}{2} \right) = - \sqrt{\frac{1 + \frac{3}{4}}{2}} \][/tex]

Simplify the expression inside the square root:
[tex]\[ \cos \left( \frac{t}{2} \right) = - \sqrt{\frac{1 + 3/4}{2}} = - \sqrt{\frac{4/4 + 3/4}{2}} = - \sqrt{\frac{7/4}{2}} = - \sqrt{\frac{7/4}{2/1}} = - \sqrt{\frac{7/4 \cdot 1/2}} = - \sqrt{\frac{7}{8}} \][/tex]

Simplify the square root:
[tex]\[ \cos \left( \frac{t}{2} \right) = - \sqrt{\frac{7}{8}} = - \frac{\sqrt{7}}{\sqrt{8}} = - \frac{\sqrt{7}}{2\sqrt{2}} = - \frac{\sqrt{7}}{2} \cdot \frac{1}{\sqrt{2}} = - \frac{\sqrt{7}}{2} \cdot \frac{\sqrt{2}}{2} = - \frac{\sqrt{7 \cdot 2}}{4} = - \frac{\sqrt{14}}{4} \][/tex]

Thus, the exact value of \(\cos \left( \frac{t}{2} \right)\) is:
[tex]\[ \cos \left( \frac{t}{2} \right) = - \frac{\sqrt{14}}{4} \][/tex]