Question

29.) evaluate the following exactly with your calculator to the nearest tenth.
a.) \\( \sin ( - 222 ^ { \circ } ) = \\)____
b.) \\( \sec ( 222 ) = \\)____
c.) \\( \tan ( 3.14 ^ { \circ } ) = \\)____
d.) \\( \csc ( \frac { 7 \pi } { 12 } ) = \\)____

Explanation:

Step1: Calculate \( \sin(-222^{\circ})\)

Use the property \( \sin(-\alpha)=-\sin\alpha\), so \( \sin(-222^{\circ})=-\sin(222^{\circ})\).
Since \( 222^{\circ}=180^{\circ} + 42^{\circ}\), and \( \sin(180^{\circ}+\theta)=-\sin\theta\) for \( \theta = 42^{\circ}\), then \( \sin(-222^{\circ})=\sin(42^{\circ})\).
Using a calculator, \( \sin(42^{\circ})\approx0.67\approx0.7\).

Step2: Calculate \( \sec(222^{\circ})\)

Recall that \( \sec\alpha=\frac{1}{\cos\alpha}\).
\( 222^{\circ}=180^{\circ}+42^{\circ}\), and \( \cos(180^{\circ}+\theta)=-\cos\theta\) for \( \theta = 42^{\circ}\), so \( \cos(222^{\circ})=-\cos(42^{\circ})\approx - 0.74\).
Then \( \sec(222^{\circ})=\frac{1}{\cos(222^{\circ})}\approx\frac{1}{- 0.74}\approx - 1.4\).

Step3: Calculate \( \tan(3.14^{\circ})\)

Using a calculator directly, \( \tan(3.14^{\circ})\approx0.05\approx0.1\).

Step4: Calculate \( \csc(\frac{7\pi}{12})\)

Recall that \( \csc\alpha=\frac{1}{\sin\alpha}\).
\( \sin(\frac{7\pi}{12})=\sin(\frac{\pi}{4}+\frac{\pi}{3})\).
Using the sum formula \( \sin(A + B)=\sin A\cos B+\cos A\sin B\), where \( A=\frac{\pi}{4}\), \( B=\frac{\pi}{3}\).
\( \sin(\frac{\pi}{4}+\frac{\pi}{3})=\sin\frac{\pi}{4}\cos\frac{\pi}{3}+\cos\frac{\pi}{4}\sin\frac{\pi}{3}=\frac{\sqrt{2}}{2}\times\frac{1}{2}+\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}=\frac{\sqrt{2}(1 + \sqrt{3})}{4}\approx0.97\).
Then \( \csc(\frac{7\pi}{12})=\frac{1}{\sin(\frac{7\pi}{12})}\approx1.0\).

Answer:

a. \(0.7\)
b. \(-1.4\)
c. \(0.1\)
d. \(1.0\)