Question

Statement: If \[x = r\sin A\cos C,y = r\sin A\sin C\] and \[z = r\cos A\], the \[{r^2} = {x^2} + {y^2} + {z^2}\].
State whether the given statement is
A. True
B. False

Answer 1

Hint: In this question, first of all write the given data and substitute these values in the given equation. Then take the common terms and use the trigonometry formula to prove the statement is true. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Given that \[x = r\sin A\cos C,y = r\sin A\sin C\] and \[z = r\cos A\]
Here we have to prove \[{r^2} = {x^2} + {y^2} + {z^2}\].
Now consider the value \[{x^2} + {y^2} + {z^2}\]
\[
   \Rightarrow {x^2} + {y^2} + {z^2} = {\left( {r\sin A\cos C} \right)^2} + {\left( {r\sin A\sin C} \right)^2} + {\left( {r\cos A} \right)^2} \\
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}{\sin ^2}A{\cos ^2}C + {r^2}{\sin ^2}A{\sin ^2}C + {r^2}{\cos ^2}A \\
\]
Taking the terms common and using the formula \[{\sin ^2}x + {\cos ^2}x = 1\], we get
\[
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}{\sin ^2}A\left( {{{\cos }^2}C + {{\sin }^2}C} \right) + {r^2}{\cos ^2}A \\
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}{\sin ^2}A\left( 1 \right) + {r^2}{\cos ^2}A{\text{ }}\left[ {\because {{\sin }^2}x + {{\cos }^2}x = 1} \right] \\
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}{\sin ^2}A + {r^2}{\cos ^2}A \\
\]
Again, taking the terms common and using the formula \[{\sin ^2}x + {\cos ^2}x = 1\], we get
\[
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}\left( {{{\sin }^2}A + {{\cos }^2}A} \right) \\
   \Rightarrow {x^2} + {y^2} + {z^2} = {r^2}{\text{ }}\left[ {\because {{\sin }^2}A + {{\cos }^2}A = 1} \right] \\
  \therefore {x^2} + {y^2} + {z^2} = {r^2} \\
\]
Hence proved that \[{x^2} + {y^2} + {z^2} = {r^2}\]
Therefore, the given statement is true
Thus, the correct option is A. True

Note: Here we have used the formula \[{\sin ^2}x + {\cos ^2}x = 1\]. To solve these kinds of problems, substitute the given values in the equation and find whether it satisfies or not. If it satisfies then the given statement is true otherwise false.