Question

How do you find the value of other trigonometric functions of theta from the information given $\cos\ \theta = - \dfrac{5}{8}$, $\tan\ \theta < 0$ ?

Answer 1

In this question, we need to find the value of the trigonometric function. Trigonometrically, there are six trigonometric functions. They are sine, cosine, tangent, secant, cosecant and cotangent. We were given that $\cos\ \theta = - \dfrac{5}{8}$ . By using the identity $\sin^{2}\theta + \cos^{2}\theta = 1$ , we can find the value of $\sin\ \theta$ . Then by taking the reciprocal of $\cos\ \theta$ , we get $\text{sec}\ \theta$ . Similarly, by taking the reciprocal of $\sin\ \theta$, we get $\text{cosec}\ \theta$. Also $\tan\ \theta$ and $\cot\ \theta$ can be found from $\sin\ \theta$ and $\cos\ \theta$.

Complete step-by-step answer:
Given that $\cos\ \theta = - \dfrac{5}{8}$
Here we need to find the value of other trigonometric functions.
First we can use the identity
$\sin^{2}\theta + \cos^{2}\theta = 1$,
On subtracting $\cos^{2}\theta$ both sides,
We get,
$\Rightarrow \ \sin^{2}\theta = 1 – \cos^{2}\theta$
Now on substituting the value of $\cos\ \theta$ ,
We get,
$\Rightarrow \ \sin^{2}\theta = 1 - \left( - \dfrac{5}{8} \right)^{2}$
On simplifying,
We get,
$\Rightarrow \ \sin^{2}\theta = 1 - \left( \dfrac{25}{64} \right)$
On taking LCM,
We get,
$\Rightarrow \ \sin^{2}\theta = \dfrac{64 – 25}{64}$
On simplifying,
We get,
$\Rightarrow \ \sin^{2}\theta = \dfrac{39}{64}$
On further simplifying,
We get,
$\Rightarrow \ \sin^{2}\theta = 0.61$
Now on taking square root on both sides,
We get,
$\sin\ \theta = \sqrt{0.61}$
$\Rightarrow \ \sin\ \theta = 0.\ 78$
Thus we get the value of sine function.
Now we know that the reciprocal of sine function is known as the cosecant function
$\Rightarrow \ \text{cosec}\ \theta = \dfrac{1}{\sin\ \theta}$
On substituting the value of $\sin\ \theta$ ,
We get,
$\Rightarrow \ \text{cosec}\ \theta = \dfrac{1}{0.78}$
On simplifying,
We get,
$\text{cosec}\ \theta = 1.28$
We also know that the reciprocal of the cosine function is known as the secant function.
$\Rightarrow \ \text{sec}\ \theta = \dfrac{1}{\cos\ \theta}$
Now on substituting the value of $\cos\ \theta$ ,
We get,
$\text{sec}\ \theta = \dfrac{1}{( - \dfrac{5}{8})}$
On taking reciprocal,
We get,
$\text{sec}\ \theta = - \dfrac{8}{5}$
On simplifying
We get,
$\text{sec}\ \theta = - 1.60$
Now we can find the value of the tangent function.
Given that $\cos\ \theta = - \dfrac{5}{8}$
On simplifying,
We get,
$\cos\ \theta = - 0.625$
We know that the ratio of sine function to the cosine function is known as tangent function.
$\Rightarrow \ \tan\ \theta = \dfrac{\sin\ \theta}{\cos\ \theta}$
Now on substituting the value of $\sin\ \theta$ and $\cos\ \theta$
We get,
$\tan\ \theta = \dfrac{0.78}{- 0.625}$
On simplifying,
We get,
$\tan\ \theta = - 1.25$
Where $\tan\ \theta < 0$
We also know that the reciprocal of tangent function is known as the cotangent function.
$\Rightarrow \ \cot\ \theta = \dfrac{1}{\tan\ \theta}$
On substituting the value of $\tan\ \theta$ ,
We get,
$\text{cot}\ \theta = \dfrac{1}{- 1.25}$
On simplifying,
We get,
$\text{cot}\ \theta = - 0.80$
Thus we get the values of all the trigonometric functions .
Final answer :
The values of $\sin\ \theta = 0.78$ , $\text{cosec}\ \theta = 1.28$ , $\text{sec}\ \theta = - 1.60$ , $\tan\ \theta = - 1.25$ and $\cot\ \theta = - 0.80$ .

Note: The concept used to solve the given problem is trigonometric functions and ratios. In order to solve these types of questions, we should have a strong grip over the trigonometric identities . Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. Trigonometrically, sine , cosine and tangent are known as the basic functions which we will find on most calculators. The other three functions are secant, cosecant and cotangent which of these are not in usual calculators.