Hello!
The answer is:
[tex]Tan(\beta)=Tan(53.13)=1.33\°[/tex]
or
[tex]Tan(\beta)=\frac{4}{3}[/tex]
Why?
To find the answer, we first need to find the value of the given angle, we can calculate it since we already know the cosine of that angle.
So, solving we have:
[tex]Cosine(\beta)=\frac{3}{5} \\\\Cosine((\beta)^{-1}=Arccos(\frac{3}{5})\\\\\beta =53.13\°[/tex]
Now, that we know that the angle is equal to 53.13°, we can calculate the value of the tangent, so:
[tex]Tan(\beta)=Tan(53.13)=1.33\°[/tex]
Also, we can find it using the Pythagorean Theorem and the trigonometric identities:
We have that:
[tex]Cosine=\frac{Adjacent}{Hypothenuse}[/tex]
So, using the given information we have:
[tex]Cosine=\frac{3}{5}[/tex]
Where,
Hypothenuse, is equal to 5.
Adjacent, side is equal to 3.
So, we are looking for the opposite side.
Then, substituting it into the Pythagorean Theorem formula, we have:
[tex]Hypothenuse^{2}=Adjacent^{2}+Opposite^{2}\\\\5^{2}=3^{2}+Opposite^{2}\\\\Opposite^{2}=25-9=16\\\\Opposite=\sqrt{16}=4[/tex]
Now that we already know the opposite side, we can find the value of the tangent that will be:
[tex]Tan(\beta)=\frac{Opposite}{Adjacent}=\frac{4}{3}[/tex]
Have a nice day!
