Question

(b)
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Explanation:

Step1: Calculate the radius r

For a point (x,y)=(2, - 1) on the terminal - side of the angle θ, use the formula \(r=\sqrt{x^{2}+y^{2}}\). So, \(r = \sqrt{2^{2}+(-1)^{2}}=\sqrt{4 + 1}=\sqrt{5}\).

Step2: Find \(\sin\theta\)

By the definition \(\sin\theta=\frac{y}{r}\), substituting \(y=-1\) and \(r = \sqrt{5}\), we get \(\sin\theta=-\frac{1}{\sqrt{5}}=-\frac{\sqrt{5}}{5}\).

Step3: Find \(\cos\theta\)

By the definition \(\cos\theta=\frac{x}{r}\), substituting \(x = 2\) and \(r=\sqrt{5}\), we get \(\cos\theta=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}\).

Step4: Find \(\tan\theta\)

By the definition \(\tan\theta=\frac{y}{x}\), substituting \(x = 2\) and \(y=-1\), we get \(\tan\theta=-\frac{1}{2}\).

Step5: Find \(\csc\theta\)

Since \(\csc\theta=\frac{1}{\sin\theta}\), substituting \(\sin\theta=-\frac{\sqrt{5}}{5}\), we get \(\csc\theta=-\sqrt{5}\).

Step6: Find \(\sec\theta\)

Since \(\sec\theta=\frac{1}{\cos\theta}\), substituting \(\cos\theta=\frac{2\sqrt{5}}{5}\), we get \(\sec\theta=\frac{\sqrt{5}}{2}\).

Step7: Find \(\cot\theta\)

Since \(\cot\theta=\frac{1}{\tan\theta}\), substituting \(\tan\theta=-\frac{1}{2}\), we get \(\cot\theta=-2\).

Answer:

\(\sin\theta=-\frac{\sqrt{5}}{5}\), \(\cos\theta=\frac{2\sqrt{5}}{5}\), \(\tan\theta=-\frac{1}{2}\), \(\csc\theta=-\sqrt{5}\), \(\sec\theta=\frac{\sqrt{5}}{2}\), \(\cot\theta=-2\)