Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator. 300°. select the correct choice below and fill in any answer boxes within your choice. cos 300° = 1/2. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined. b. select the correct choice below and fill in any answer boxes within your choice. tan 300° = -√3. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined. b. select the correct choice below and fill in any answer boxes within your choice. csc 300° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined. b.

Explanation:

Step1: Recall the reference - angle

The angle \(300^{\circ}\) is in the fourth - quadrant. The reference angle \(\theta_{r}=360^{\circ}-300^{\circ} = 60^{\circ}\).

Step2: Find the cosine value

In the fourth - quadrant, \(\cos\theta>0\). Since \(\cos300^{\circ}=\cos(360^{\circ} - 60^{\circ})\), and \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) with \(A = 360^{\circ}\), \(B = 60^{\circ}\), \(\cos360^{\circ}=1\), \(\sin360^{\circ}=0\), \(\cos60^{\circ}=\frac{1}{2}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\cos300^{\circ}=\cos60^{\circ}=\frac{1}{2}\).

Step3: Find the tangent value

In the fourth - quadrant, \(\tan\theta<0\). \(\tan300^{\circ}=\tan(360^{\circ}-60^{\circ})=-\tan60^{\circ}=-\sqrt{3}\).

Step4: Find the cosecant value

First, \(\sin300^{\circ}=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}\) (because sine is negative in the fourth - quadrant). Since \(\csc\theta=\frac{1}{\sin\theta}\), then \(\csc300^{\circ}=\frac{1}{\sin300^{\circ}}=-\frac{2\sqrt{3}}{3}\).

Answer:

\(\cos300^{\circ}=\frac{1}{2}\), \(\tan300^{\circ}=-\sqrt{3}\), \(\csc300^{\circ}=-\frac{2\sqrt{3}}{3}\)