Question

What trigonometric ratios from ${0^ \circ }$ to ${90^ \circ }$ are equal to $1$ ?

Answer 1

Hint: We know the values of basic trigonometric ratios in the given range. So, we will check each ratio one by one and try to find the values which are equal to one. We know that there are three basic trigonometric ratios and remaining are the reciprocals.

Complete step-by-step answer:
We have given the interval ${0^ \circ }$ to ${90^ \circ }$. In this interval we have to find the angles for which the value of the trigonometric ratio is $1$ .
We will start with the basic ratio of $\sin x$ .
We know that the ratio is defined as the ratio of the opposite side to the hypotenuse.
The value of the sine ratio for the angle ${90^ \circ }$ is one.
Hence, we know that sine ratio takes value one in the given interval.
Therefore, the reciprocal will also take the value one at the same point.
Therefore, cosec will also take the value one at ${90^ \circ }$.
Now we will move to the next ratio that is $\cos x$.
We know that it is defined as the ratio of the adjacent side to the hypotenuse.
The value of cosine ratio is one when the angle is ${0^ \circ }$.
Therefore, its reciprocal will also take the same value at the same point.
This implies that secant will also be one when the angle is ${0^ \circ }$.
Finally, we will study the tangent function.
We know that tangent function is defined as the ratio of sine and cosine function.
Therefore, the value of tangent function will be one when both the functions are equal.
We know that $\sin x = \cos x = \dfrac{1}{{\sqrt 2 }}$ when $x = {45^ \circ }$ .
Therefore, the tangent function will be one when the angle is ${45^ \circ }$ .
Similarly, the cotangent function will also take the same value for the same point.
That means the cotangent function will also take the value one at angle $x = {45^ \circ }$ .
Thus, we conclude that each trigonometric function takes the value one at some point in the given interval.

Note: It is enough to find the ratios for which the value is one. As the reciprocal of one is also the reciprocals have to follow the same values. It is also important to note that in the given interval if a function takes the value one once then it cannot take the same value again in that interval as the period of each function is greater than the given interval. We don’t have to check for every value.