Introduction to Trigonometry
CH08 Question Bank
CBSE_CH08_QUESTION_BANK
Grade 10
Question:
If $x=r\sin A\cos B$, $y=r\sin A\sin B$ and $z=r\cos A$, prove that $x^2+y^2+z^2=r^2$.
Step-by-Step Solution
Key Concept: Square and add all three expressions, then factor out common terms and apply the Pythagorean identity twice.
$x^2+y^2=r^2\sin^2A\cos^2B+r^2\sin^2A\sin^2B=r^2\sin^2A(\cos^2B+\sin^2B)$. [1.5 Marks]
Using $\sin^2B+\cos^2B=1$: $x^2+y^2=r^2\sin^2A(1)=r^2\sin^2A$. [1.0 Mark]
$x^2+y^2+z^2=r^2\sin^2A+r^2\cos^2A=r^2(\sin^2A+\cos^2A)$. [1.5 Marks]
Using $\sin^2A+\cos^2A=1$: $x^2+y^2+z^2=r^2(1)=r^2$. Hence proved. [1.0 Mark]
Correct Answer:
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