If \(\cos x = \frac{5}{{13}}\) , x lies in the fourth quadrant, and determines the value of the other five trigonometric functions?
To determine the values of the other five trigonometric functions when \(\cos x = \frac{5}{{13}}\) and \(x\) lies in the fourth quadrant, we will use fundamental trigonometric identities and rules specific to quadrants.
Understanding Quadrants and Signs:
- In the fourth quadrant, the angle \(x\) is between \(270^\circ\) and \(360^\circ\) (or \(\frac{3\pi}{2}\) and \(2\pi\) radians).
- In this quadrant, the cosine function (\(\cos x\)) and its reciprocal, the secant function (\(\sec x\)), are positive.
- All other trigonometric functions, namely sine (\(\sin x\)), tangent (\(\tan x\)), cosecant (\(\csc x\)), and cotangent (\(\cot x\)), are negative in the fourth quadrant.
Trigonometric Functions Calculation
Given: \(\cos x = \frac{5}{{13}}\)
Sine Function: Determining \(\sin x\)
We use the fundamental trigonometric identity: \(\sin^2 x + \cos^2 x = 1\)
Substitute the given value of \(\cos x\):
\[\sin^2 x + \left(\frac{5}{13}\right)^2 = 1\] \[\sin^2 x + \frac{25}{169} = 1\] \[\sin^2 x = 1 - \frac{25}{169}\] \[\sin^2 x = \frac{169 - 25}{169}\] \[\sin^2 x = \frac{144}{169}\]Taking the square root of both sides:
\[\sin x = \pm \sqrt{\frac{144}{169}}\] \[\sin x = \pm \frac{12}{13}\]Since \(x\) lies in the fourth quadrant, the value of \(\sin x\) must be negative.
Therefore, \[\sin x = -\frac{12}{13}\]
Secant Function: Determining \(\sec x\)
The secant function is the reciprocal of the cosine function:
\[\sec x = \frac{1}{\cos x}\]Substitute the given value of \(\cos x\):
\[\sec x = \frac{1}{\frac{5}{13}}\]Therefore, \[\sec x = \frac{13}{5}\]
This result is positive, which is consistent with the rules for the fourth quadrant.
Tangent Function: Determining \(\tan x\)
The tangent function is the ratio of sine to cosine:
\[\tan x = \frac{\sin x}{\cos x}\]Substitute the values of \(\sin x\) and \(\cos x\):
\[\tan x = \frac{-\frac{12}{13}}{\frac{5}{13}}\] \[\tan x = -\frac{12}{5}\]This result is negative, which is consistent with the rules for the fourth quadrant.
Cosecant Function: Determining \(\csc x\)
The cosecant function is the reciprocal of the sine function:
\[\csc x = \frac{1}{\sin x}\]Substitute the value of \(\sin x\):
\[\csc x = \frac{1}{-\frac{12}{13}}\]Therefore, \[\csc x = -\frac{13}{12}\]
This result is negative, which is consistent with the rules for the fourth quadrant.
Cotangent Function: Determining \(\cot x\)
The cotangent function is the reciprocal of the tangent function:
\[\cot x = \frac{1}{\tan x}\]Substitute the value of \(\tan x\):
\[\cot x = \frac{1}{-\frac{12}{5}}\]Therefore, \[\cot x = -\frac{5}{12}\]
This result is negative, which is consistent with the rules for the fourth quadrant.
Summary of Trigonometric Values
Here is a summary of all six trigonometric function values for the given angle \(x\):
| Function | Value |
|---|---|
| \(\cos x\) | \(\frac{5}{13}\) |
| \(\sin x\) | \(-\frac{12}{13}\) |
| \(\sec x\) | \(\frac{13}{5}\) |
| \(\csc x\) | \(-\frac{13}{12}\) |
| \(\tan x\) | \(-\frac{12}{5}\) |
| \(\cot x\) | \(-\frac{5}{12}\) |
Based on our calculations, the values for the other five trigonometric functions are:
\(\sec x = \frac{{13}}{5},\sin x =- \frac{{12}}{{13}},\rm cosec\ x =- \frac{{13}}{{12}},\tan x = -\frac{{12}}{5},\cot x =- \frac{5}{{12}}\)
Important Questions from Trigonometric Ratios
- If $\sin\theta = \frac{p^2 - 1}{p^2 + 1}$, and $\theta$ is an acute angle, then find the value of $\cos\theta$.
- Let ABC be a right-angled triangle such that $\angle B = 90^\circ$, AB = 12 cm and AC = 13 cm. Find the value of $\tan A - \cot C$.
What is \(\tan \alpha + \tan \beta\) equal to?
- If \(\cos\theta = 24/25\) then \(\cot\theta\) will be
- Value of \(\sin 60^\circ / \cos 60^\circ\)
