Question

Which trigonometric function does not use the hypotenuse in its ratio?
(a) sine
(b) cosine
(c) tangent
(d) secant

Answer 1

Hint:First of all, recall all the trigonometric ratios. Now out of those, select that trigonometric ratio where the hypotenuse is not used and mark the answer accordingly.

Complete step-by-step answer:
In this question, we have to find the trigonometric ratio which does not use the hypotenuse in its ratio out of sine, cosine, tangent, and secant.
Before proceeding with the question, let us first learn about all the trigonometric
ratios. Basically, we write all the trigonometric ratios with respect to a right triangle. It is a type of triangle that has one angle that measures \[{{90}^{o}}\]. Given below is a right triangle ABC right angled at C and having sides as a, b and c.

We use trigonometry to find angles and sides of the right triangle. The various functions in trigonometry are sine, cosine, tangent, cosecant, secant, and cotangent. They are simply one side of a right-angled triangle divided by another. Let us take angle A of the above triangle, then side BC would be perpendicular, side AC would be base and side AB would be hypotenuse, so we can write,
\[\sin A=\dfrac{perpendicular}{hypotenuse}=\dfrac{a}{c}\]
\[\operatorname{cosec}A=\dfrac{hypotenuse}{perpendicular}=\dfrac{c}{a}\]
\[\cos A=\dfrac{base}{hypotenuse}=\dfrac{b}{c}\]
\[\sec A=\dfrac{hypotenuse}{base}=\dfrac{c}{b}\]
\[\tan A=\dfrac{perpendicular}{base}=\dfrac{a}{b}\]
\[\cot A=\dfrac{base}{perpendicular}=\dfrac{b}{a}\]
Now let us consider our question. We can see that in trigonometric ratio sine, we use perpendicular and hypotenuse with respect to the given angle in the triangle. In trigonometric ratio cosine, we use base and hypotenuse with respect to the given angle in the triangle. In trigonometric ratio tangent, we use perpendicular and base with respect to the given angle in the triangle. In trigonometric ratio secant, we use hypotenuse and base with respect to the given angle in the triangle. So, out of sine, cosine, tangent, and secant, the tangent is the trigonometric ratio that does not use the hypotenuse in its ratio.
Hence, option c is correct.

Note: First of all, students must have knowledge of and remember all trigonometric ratios clearly apart from memorizing the formulas of trigonometry because this is the most basic and very useful thing while solving questions related to trigonometry. Also, students should properly know to use the Pythagoras theorem to use these ratios easily.