Question

The value of \[\cos {{10}^{\circ }}-\sin {{10}^{\circ }}\] is?
a.Positive
b.Negative
c.0
d.1

Answer 1

Hint: Mathematics includes the study of topics which are related to quantity, structure, space and change. It has no generally accepted definition. It involves various operations which are performed using various operators such as addition multiplication division and subtraction. By the combination of these primary operators, trigonometric expression is obtained for a triangle. Using these properties, we can solve the above problem.

Complete step-by-step answer:
Mathematics is related to all the phenomena occurring in the world. When mathematical structures are good models of real phenomena mathematical reasoning can be used to provide insight or predictions about nature. One of the important sub branches of mathematics is trigonometry which involves the study of angles.

Let the value of $\sin {{10}^{\circ }}-\cos {{10}^{\circ }}$be y.

y= $\sin {{10}^{\circ }}-\cos {{10}^{\circ }}$.

For any value of $\theta $, which is equal to or less than $\dfrac{\pi }{4}$ and greater than 0.

This can be expressed mathematically as: $0\le \theta <\dfrac{\pi }{4}$

This implies that for this case the value of cosine is greater than value of sine.
$\therefore \cos \theta >\sin \theta $

Therefore, $\cos {{10}^{\circ }}>\sin {{10}^{\circ }}$.

Hence, on subtraction of both the values, the value of y is positive.

Hence, option (a) is correct.

Note: The key step for solving this problem is observing the pattern of the two trigonometric functions as the value of angle progresses. Once the same trigonometric trend is observed then evaluation for the angle can be done. This problem can also be solved alternatively by transforming the angle into cosine or sine by using the identity sin (90° - θ) = cos θ. Hence, when both values are in one variable then calculation can be done easily.