Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>If \(x^2 + y^2 + z^2 = r^2\), then \(\tan^{-1}\left(\frac{xy}{zr}\right) + \tan^{-1}\left(\frac{yz}{xr}\right) + \tan^{-1}\left(\frac{xz}{yr}\right)\) is equal to</p>
<p>(a) \(\pi\)</p>
<p>(b) \(\frac{\pi}{2}\)</p>
<p>(c) \(0\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use parametrization of the sphere constraint and inverse tangent addition formulas
<p>Let $x = r\sin\alpha\cos\beta$, $y = r\sin\alpha\sin\beta$, $z = r\cos\alpha$. Then the sum evaluates to 0.</p>
Correct Answer: C