Question

use the unit circle to find the exact value of each of the following. 1. sin\\(\frac{3\pi}{2}\\) 2. csc\\(\frac{7\pi}{4}\\) =\\(\frac{2}{\sqrt{2}}=\sqrt{2}\\) 3. tan\\(\frac{\pi}{3}\\) 4. sec\\(\left(-\frac{5\pi}{6}\
ight)\\) 5. cot\\(\left(-\frac{11\pi}{6}\
ight)\\) 6. cos\\(\left(-\frac{\pi}{3}\
ight)\\) 7. csc\\(\frac{26\pi}{3}\\) 8. tan\\(\left(-\frac{10\pi}{3}\
ight)\\) 9. sec\\(\frac{19\pi}{4}\\) 10. cos\\((-225^{\circ})\\) 11. cos 0 12. sec\\((-780^{\circ})\\) 13. csc\\(\frac{2\pi}{3}\\) 14. cot\\(\frac{4\pi}{3}\\) 15. sin\\((-480^{\circ})\\) 16. cos\\(\left(-\frac{13\pi}{4}\
ight)\\) 17. sin\\(\frac{47\pi}{6}\\) 18. sec 900^{\circ}

Explanation:

Step1: Find the reference angle

For \(\cos(-225^{\circ})\), use the property \(\cos(-\theta)=\cos\theta\), so \(\cos(-225^{\circ})=\cos225^{\circ}\).
\(225^{\circ}=180^{\circ} + 45^{\circ}\), the reference angle \(\theta'=225^{\circ}-180^{\circ}=45^{\circ}\).
Since \(225^{\circ}\) is in the third - quadrant and \(\cos\theta\) is negative in the third - quadrant.

Step2: Calculate the value

\(\cos225^{\circ}=-\cos45^{\circ}\).
We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), so \(\cos(-225^{\circ})=-\frac{\sqrt{2}}{2}\).

Answer:

\(-\frac{\sqrt{2}}{2}\)