Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>The value of the expression \(\sin\left(2\tan^{-1}\frac{1}{3}\right) + \cos\left(\tan^{-1}2\sqrt{2}\right)\) is</p>
<p>(a) \(\frac{13}{14}\)</p>
<p>(b) \(\frac{14}{13}\)</p>
<p>(c) Cannot be determined</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Convert inverse trigonometric expressions to direct trigonometric values using right triangle relationships
<p>For $\sin(2\tan^{-1}\frac{1}{3})$: Let $\tan^{-1}\frac{1}{3} = \alpha$. Then $\sin(2\alpha) = \frac{2\tan\alpha}{1+\tan^2\alpha} = \frac{2 \cdot \frac{1}{3}}{1 + \frac{1}{9}} = \frac{3}{5}$. For $\cos(\tan^{-1}2\sqrt{2})$: Let $\tan^{-1}2\sqrt{2} = \beta$. Then $\cos\beta = \frac{1}{\sqrt{1+(2\sqrt{2})^2}} = \frac{1}{3}$. Sum = $\frac{3}{5} + \frac{1}{3} = \frac{14}{15}$.</p>
Correct Answer: A