The exact values of the cotangent and the cosine of the angle are [tex]\frac{\sqrt{119}}{5}[/tex] and [tex]\frac{\sqrt{119}}{12}[/tex], respectively.
According to the information given, the angle resides in the first quadrant of the Cartesian plane. Hence, the cosine and the cotangent of the angle are also positive.
By trigonometry we remember the concepts of sine, cotangent and cosine:
[tex]\sin \theta = \frac{y}{r}[/tex] (1)
[tex]\cos \theta = \frac{x}{r}[/tex] (2)
[tex]\cot \theta = \frac{x}{y}[/tex] (3)
Where [tex]r = \sqrt{x^{2}+y^{2}}[/tex].
If we know that [tex]y = 5[/tex] and [tex]r = 12[/tex], then the cotangent and the cosine of the angle are, respectively:
[tex]x = \sqrt{r^{2}-y^{2}}[/tex]
[tex]x = \sqrt{12^{2}-5^{2}}[/tex]
[tex]x = \sqrt{119}[/tex]
[tex]\cos \theta = \frac{\sqrt{119}}{12}[/tex]
[tex]\cot \theta = \frac{\sqrt{119}}{5}[/tex]
The exact values of the cotangent and the cosine of the angle are [tex]\frac{\sqrt{119}}{5}[/tex] and [tex]\frac{\sqrt{119}}{12}[/tex], respectively.
We kindly invite to check this question on trigonometric functions: https://brainly.com/question/6904750
