Question

The value of $\sin A$ or $\cos A$ never exceeds the value_____.

Answer 1

Hint: First use the trigonometric ratio to find the ratio of sine and cosine in terms of perpendicular, base and hypotenuse. Then use the Pythagoras theorem to get the result that the Hypotenuse is the longest side of the right triangle. Use this result to get the desired result.

Complete step-by-step answer:
For any right angle triangle, which is right-angled at angle B. Then for the angle A, the base of the triangle is AB, perpendicular to the triangle is BC, and the hypotenuse of the triangle is AC. The figure of the triangle is shown.

Using the trigonometric ratios, we know that the sine of the angle A is the ratio of the perpendicular and the hypotenuse and cosine is the ratio of the base and the hypotenuse. That is,
$\sin A = \dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}$
$\cos A = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}}$
Using the Pythagoras theorem, the square of the hypotenuse of the triangle is equal to the sum of the square of the perpendicular and the base of the triangle.
${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}$
Then we can conclude that:
${\left( {{\text{Hypotenuse}}} \right)^2} > {\left( {{\text{Perpendicular}}} \right)^2}$and${\left( {{\text{Hypotenuse}}} \right)^2} > {\left( {{\text{Base}}} \right)^2}$
$ \Rightarrow {\text{Hypotenuse}} > {\text{Perpendicular}}$ and ${\text{Hypotenuse}} > {\text{Base}}$
So, we get the conclusion that the hypotenuse of the right triangle is the longest side of the triangle.
Thus, the ratio of sine and cosine is always less than 1.
So, the value of $\sin A$ or $\cos A$ never exceeds the value 1.

Note: While using the Pythagoras theorem, remember that the perpendicular and the base of the triangle are changed with the angle. We have used angle A for finding the ratio of the sine and cosine. In case, if we use the angle C, then the perpendicular is taken as AB and base is taken as BC.