Question

The value of $\cos {10^0} - \sin {10^0}$ is
$
  {\text{A}}{\text{. positive}} \\
  {\text{B}}{\text{. negative}} \\
  {\text{C}}{\text{. 0}} \\
  {\text{D}}{\text{. 1}} \\
 $

Answer 1

Hint: To find the value of a given question either we have remembered these values then we can check or we have knowledge of the trend of these functions that either they are increasing or decreasing then we can compare them.

Complete step-by-step answer:

We have $\cos {10^0} - \sin {10^0}$
As we know value of $\cos {0^0} = 1$ and $\cos {30^0} = \dfrac{{\sqrt 3 }}{2}$ here we can clearly see cosine function is decreasing from zero degree to ninety degree and value at zero degree is maximum which is 1.
As we can see value of $\cos {30^0} = \dfrac{{\sqrt 3 }}{2}$ that means value of $\cos {10^0}$ will lie between 1 and $\dfrac{{\sqrt 3 }}{2}$
That means $\dfrac{{\sqrt 3 }}{2} \leqslant \cos {10^0} \leqslant 1$ $ \ldots \ldots \left( i \right)$
Now
As we know value of $\sin {0^0} = 0$ and $\sin {30^0} = \dfrac{1}{2}$ here we can clearly see sine function is increasing from zero degree to ninety degree and value at zero is minimum which is 0.
And value of $\sin {30^0} = \dfrac{1}{2}$ that means $\sin {10^0}$ will lie between 0 and $\dfrac{1}{2}$
That is $0 \leqslant \sin {10^0} \leqslant \dfrac{1}{2}$ $ \ldots \ldots \left( {ii} \right)$
Here we can see from equation (i) and (ii) $\cos {10^0}$ is greater than $\sin {10^0}$
So result of subtraction will be positive and hence,

Option A is the correct option.

Note: Whenever we get this type of question the key concept of solving is described in the solution or we may directly answer if we know $\cos {10^0} = 0.94$ and $\sin {10^0} = 0.17$ then answer is very easily known but these values are not remembered so care on above solution for this type of questions.