Answer:
[tex]sinx=-\dfrac{12}{13}[/tex]
[tex]cosx=-\dfrac{5}{13}[/tex]
[tex]cotx=\dfrac{5}{12}[/tex]
Step-by-step explanation:
Given that:
[tex]\dfrac{12}{5} = tan(x)[/tex]
[tex]\pi <x < 3\pi/2[/tex]
i.e. x is in 3rd quadrant. So tan is positive.
To find:
sin(x), cos(x), and cot(x).
Solution:
Given that:
[tex]\dfrac{12}{5} = tan(x)[/tex]
We know by trigonometric identities that:
[tex]tan\theta =\dfrac{Perpendicular}{Base}[/tex]
Comparing with the given values:
[tex]\theta=x[/tex]
Perpendicular = 12 units
Base = 5 units
Using pythagorean theorem, we can find out hypotenuse:
According to pythagorean theorem:
[tex]\text{Hypotenuse}^{2} = \text{Base}^{2} + \text{Perpendicular}^{2}[/tex]
[tex]\Rightarrow Hypotenuse=\sqrt{12^2+5^2}\\\Rightarrow Hypotenuse=\sqrt{169} = 13 units[/tex]
We can easily find out the values of:
[tex]sinx, cos x\ and\ cot x[/tex]
[tex]sin\theta =\dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]sinx =\dfrac{12}{13}[/tex]
Given that x is in 3rd quadrant, sinx will be negative.
[tex]\therefore sinx =-\dfrac{12}{13}[/tex]
[tex]sin\theta =\dfrac{Base}{Hypotenuse}[/tex]
[tex]cosx =\dfrac{5}{13}[/tex]
Given that x is in 3rd quadrant, cosx will be negative.
[tex]\therefore cosx =-\dfrac{5}{13}[/tex]
[tex]cot\theta = \dfrac{1}{tan\theta}[/tex]
Given that x is in 3rd quadrant, cotx will be positive.
[tex]cotx = \dfrac{1}{\dfrac{12}{5}} = \dfrac{5}{12}[/tex]
