Question

If $\cos A=\dfrac{4}{5}$ , find the value of other trigonometric ratios.

Answer 1

We have given the value of $\cos A$ so from $\cos A$ we can find other trigonometric ratios such as$\sin A,\tan A,\cot A,\sec A\And \cos ecA$ . We know that $\cos A=\dfrac{B}{H}$ from this equation we can find the perpendicular of the triangle corresponding to angle A using Pythagoras theorem. Now, we have all the sides so we can easily find the other trigonometric ratios corresponding to angle A.

Complete step-by-step answer:
The value of $\cos A$ given in the above question is:
$\cos A=\dfrac{4}{5}$
The below figure is showing a right triangle ABC right angled at B.

In the above figure, “P” stands for perpendicular with respect to angle A, “B” stands for the base of a triangle with respect to angle A and “H” stands for the hypotenuse of the triangle with respect to angle A.
We know from the trigonometric ratio that:
$\cos A=\dfrac{B}{H}$
In the above equation, B stands for base and H stands for hypotenuse of the triangle corresponding to angle A so from the Pythagoras theorem we can find the perpendicular of the triangle and we are representing perpendicular with a symbol “P”.
$\begin{aligned} & {{H}^{2}}={{P}^{2}}+{{B}^{2}} \\\ & \Rightarrow 25={{P}^{2}}+16 \\\ & \Rightarrow {{P}^{2}}=9 \\\ & \Rightarrow P=3 \\\ \end{aligned}$
Now, we can easily find the other trigonometric ratios with respect to angle A.
We know that:
$\sin A=\dfrac{P}{H}$
Substituting the value of $P=3$ and $H=5$ we get,
$\sin A=\dfrac{3}{5}$
Now, we are going to find the trigonometric ratio $\tan A$ :
$\tan A=\dfrac{P}{B}$
Substituting the value of $P=3$ and $B=4$ in the above equation we get,
$\tan A=\dfrac{3}{4}$
We know that $\cot A$ is the reciprocal of $\tan A$ so,
$\cot A=\dfrac{4}{3}$
We know that $\cos ecA$ is the reciprocal of $\sin A$ so,
$\cos ecA=\dfrac{5}{3}$
We know that $\sec A$ is the reciprocal of $\cos A$ so,
$\sec A=\dfrac{5}{4}$
And the value of $\cos A$ is already given in the question.
Hence, we have found all the trigonometric ratios corresponding to angle A.

Note: While reading the above question you might get confused that what are the trigonometric ratios and if you could understand the trigonometric ratios you might get confused like do I have to find the trigonometric ratios for all the angles of the given triangle.
The remedy of all this confusion is that trigonometric ratios are $\sin ,\tan ,\cot ,\sec \And \cos ec$ of a particular angle and as $\cos A$ is given in the question so we have to find the trigonometric ratios corresponding to angle A.