Question

1. an angle \\( \theta \\) is spun from the positive \\( x \\)-axis counterclockwise to a terminal arm. the terminal arm contains the point \\( (3,-4 \sqrt{2}) \\) find the exact value of \\( \sec \theta \\). sketch the principal angle. c-3 marks \\( r = \sqrt { x ^ { 2 } + y ^ { 2 } } \\) \\( x = 3 \\) \\( y = - 4 \sqrt { 2 } \\) \\( y = \sqrt { 3 ^ { 2 } + ( - 4 \sqrt { 2 } ) ^ { 2 } } \\) \\( = \sqrt { 9 + 32 } = \sqrt { 41 } \\) \\( \cos \theta = \frac { x } { r } \\) \\( x = 3 \\) \\( r = \sqrt { 41 } \\) \\( \cos \theta = \frac { 3 } { \sqrt { 41 } } \\) \\( \sec \theta = \frac { 1 } { \cos \theta } \\) \\( \sec \theta = \frac { \sqrt { 41 } } { 3 } \\)

Explanation:

Step1: Calculate the value of \(r\)

Use the formula \(r = \sqrt{x^{2}+y^{2}}\). Given \(x = 3\) and \(y=-4\sqrt{2}\), then \(r=\sqrt{3^{2}+(-4\sqrt{2})^{2}}=\sqrt{9 + 32}=\sqrt{41}\).

Step2: Find the value of \(\cos\theta\)

By the definition \(\cos\theta=\frac{x}{r}\), substituting \(x = 3\) and \(r=\sqrt{41}\), we get \(\cos\theta=\frac{3}{\sqrt{41}}\).

Step3: Calculate the value of \(\sec\theta\)

Since \(\sec\theta=\frac{1}{\cos\theta}\), substituting \(\cos\theta=\frac{3}{\sqrt{41}}\), we have \(\sec\theta=\frac{\sqrt{41}}{3}\).

Answer:

\(\sec\theta=\frac{\sqrt{41}}{3}\)