solve the equation for solutions over the interval 0°, 360°).\n\\(cot 0 + 3 csc 0 = 7\\)\nselect the correct…

solve the equation for solutions over the interval 0°, 360°).\n\\(cot 0 + 3 csc 0 = 7\\)\nselect the correct choice below and, if necessary, fill in the\nanswer box to complete your choice.\na. the solution set is {}\n(type your answer in degrees. do not include the\ndegree symbol in your answer. round to one decimal\nplace as needed. use a comma to separate answers\nas needed.)\nb. the solution is the empty set

solve the equation for solutions over the interval 0°, 360°).\n\\(cot 0 + 3 csc 0 = 7\\)\nselect the correct choice below and, if necessary, fill in the\nanswer box to complete your choice.\na. the solution set is {}\n(type your answer in degrees. do not include the\ndegree symbol in your answer. round to one decimal\nplace as needed. use a comma to separate answers\nas needed.)\nb. the solution is the empty set

Answer

Explanation:

Step1: Express in terms of sine and cosine

Recall that (\cot\theta=\frac{\cos\theta}{\sin\theta}) and (\csc\theta = \frac{1}{\sin\theta}). The equation (\cot\theta+3\csc\theta = 7) becomes (\frac{\cos\theta}{\sin\theta}+\frac{3}{\sin\theta}=7). Since (\sin\theta\neq0) (because (\csc\theta) is defined), we can multiply through by (\sin\theta) to get (\cos\theta + 3=7\sin\theta).

Step2: Square both sides

Square the equation (\cos\theta+3 = 7\sin\theta). ((\cos\theta + 3)^2=(7\sin\theta)^2). Expand: (\cos^{2}\theta+6\cos\theta + 9 = 49\sin^{2}\theta). Use the identity (\sin^{2}\theta=1-\cos^{2}\theta). (\cos^{2}\theta+6\cos\theta + 9=49(1 - \cos^{2}\theta)). (\cos^{2}\theta+6\cos\theta + 9=49-49\cos^{2}\theta). (50\cos^{2}\theta+6\cos\theta - 40 = 0). Divide by (2): (25\cos^{2}\theta+3\cos\theta - 20=0). Let (x = \cos\theta), then (25x^{2}+3x - 20=0). Using the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) with (a = 25), (b = 3), (c=-20). (x=\frac{-3\pm\sqrt{9+2000}}{50}=\frac{-3\pm\sqrt{2009}}{50}). (\cos\theta=\frac{-3\pm\sqrt{2009}}{50}). (\cos\theta=\frac{-3 + 44.82}{50}\approx0.836) or (\cos\theta=\frac{-3-44.82}{50}\approx - 0.956).

Step3: Find (\theta) values

If (\cos\theta\approx0.836), then (\theta=\cos^{-1}(0.836)\approx33.3^{\circ}) or (\theta = 360 - 33.3=326.7^{\circ}). If (\cos\theta\approx - 0.956), then (\theta=\cos^{-1}(-0.956)\approx163.1^{\circ}) or (\theta = 196.9^{\circ}). Check these solutions in the original equation (\cot\theta+3\csc\theta = 7). For (\theta = 33.3^{\circ}): (\cot(33.3)=\frac{\cos(33.3)}{\sin(33.3)}\approx1.5), (\csc(33.3)=\frac{1}{\sin(33.3)}\approx1.8). (\cot(33.3)+3\csc(33.3)\approx1.5 + 3\times1.8=1.5+5.4 = 6.9\approx7). For (\theta = 163.1^{\circ}): (\cot(163.1)=\frac{\cos(163.1)}{\sin(163.1)}\approx - 3.3), (\csc(163.1)=\frac{1}{\sin(163.1)}\approx3.4). (\cot(163.1)+3\csc(163.1)\approx-3.3+3\times3.4=-3.3 + 10.2=6.9\approx7).

Answer:

(33.3,163.1,196.9,326.7)