Answer:
[tex] \cos( \theta) = \frac{5}{8} \\ {8}^{2} = {5}^{2} + {opp}^{2} \\ {opp}^{2} = 64 - 25 = 39 \\ opp = \sqrt{39} \\ \sin(\theta) = \frac{\sqrt{39}}{8} \\ \tan(\theta) = \frac{ \sqrt{39} }{5} \\ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{ \sqrt{39} }{5}} \\ \cot(\theta) = \frac{5}{ \sqrt{39} } \\ \csc( \theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{\sqrt{39}}{8}} \\ \csc( \theta) = \frac{8}{ \sqrt{39} } \\ \sec( \theta) = \frac{1}{\cos( \theta) } = \frac{1}{ \frac{5}{8} } \\ \sec( \theta) = \frac{8}{5} [/tex]
For given angle [tex]\theta[/tex],
[tex]\bold{cot(\theta)=\frac{5}{6.24}}\\\\ \bold{cos(\theta)=\frac{5}{8}}\\\\ \bold{sec(\theta)=\frac{8}{5}}[/tex]
In a right triangle, for angle Ф
cos(Ф) = adjacent side of angle Ф / hypotenuse
In a right triangle, for angle Ф
cos(Ф) = opposite side of angle Ф / hypotenuse
In a right triangle for an angle Ф,
sec(Ф) = 1 / (cos(Ф))
In a right triangle for an angle Ф,
cot(Ф) = cos(Ф) / sin(Ф)
For a right triangle,
[tex]c^2=a^2+b^2[/tex]
where 'c' represents the hypotenuse
'a' and 'b' be the other two sides of the right triangle
For given example,
The adjacent side of [tex]\theta[/tex] is 5 and the hypotenuse is 5.
[tex]\bold {cos(\theta)=\frac{5}{8}}[/tex]
We find the opposite side of the angle [tex]\theta[/tex] using Pythagoras theorem.
Let 'x' be the opposite side of the angle [tex]\theta[/tex] .
By Pythagoras theorem,
[tex]\Rightarrow 8^{2} =x^{2} + 5^{2}\\\\ \Rightarrow x^{2} =64-25\\\\ \Rightarrow x=\sqrt{39}\\\\ \Rightarrow x=6.24[/tex]
So, the opposite side of the angle [tex]\theta[/tex] is 6.24 units.
[tex]sin(\theta)=\frac{6.24}{8}[/tex]
So, the value of [tex]cot(\theta)[/tex] would be,
[tex]\Rightarrow cot(\theta)=\frac{cos(\theta)}{sin(\theta)}\\\\\Rightarrow cot(\theta)=\frac{\frac{5}{8} }{\frac{6.24}{8} } \\\\\Rightarrow \bold{cot(\theta)=\frac{5}{6.24}}[/tex]
Now we find the value of [tex]\bold{sec(\theta)}[/tex]
[tex]\Rightarrow sec(\theta)=\frac{1}{cos(\theta)}\\\\\Rightarrow sec(\theta)=\frac{1}{\frac{5}{8} }\\\\ \Rightarrow \bold{sec(\theta)=\frac{8}{5}}[/tex]
Therefore, for given angle [tex]\theta[/tex],
[tex]\bold{cot(\theta)=\frac{5}{6.24}}\\\\ \bold{cos(\theta)=\frac{5}{8}}\\\\ \bold{sec(\theta)=\frac{8}{5}}[/tex]
Learn more about cotθ, cosθ, and secθ here:
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