If $\sec\theta = \dfrac{13}{5}$, find the values of all the other five trigonometric ratios of $\theta$.
Step-by-Step Solution
Key Concept: Use the reciprocal relation to find cos θ, then the Pythagorean identity to find sin θ, and finally derive the rest.
$\cos\theta=\dfrac{1}{\sec\theta}=\dfrac{5}{13}$. [1.0 Mark]
$\sin^2\theta=1-\cos^2\theta=1-\dfrac{25}{169}=\dfrac{144}{169}\Rightarrow\sin\theta=\dfrac{12}{13}$. [1.5 Marks]
$\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{12/13}{5/13}=\dfrac{12}{5}$. [1.0 Mark]
$\text{cosec}\,\theta=\dfrac{1}{\sin\theta}=\dfrac{13}{12}$. [0.75 Mark]
$\cot\theta=\dfrac{1}{\tan\theta}=\dfrac{5}{12}$. [0.75 Mark]
Correct Answer: