{Given: cos(A)=\frac: sqrt{203}/{18}
{Find: tan(A)
Find the positive value of the above in simplest radical form.
Sorry, we do not have Latex.
So
cos(A)=sqrt(203)/18
|tan(A)|
=sqrt(sec²(A)-1)
=sqrt(18²/203 -1)
=sqrt(324/203-1)
=sqrt(121/203)
=11/sqrt(203)
To find the value of tan(A), we can use the Pythagorean identity for trigonometric functions, which states that:
sin^2(A) + cos^2(A) = 1
Since we are given the value of cos(A), we can substitute it into the equation:
sin^2(A) + \left(\frac{\sqrt{203}}{18}\right)^2 = 1
Let's solve for sin(A):
sin^2(A) + \frac{203}{18^2} = 1
sin^2(A) + \frac{203}{324} = 1
sin^2(A) = 1 - \frac{203}{324}
sin^2(A) = \frac{121}{324}
To find the value of sin(A), we need to determine whether it is positive or negative. Since A could be any angle in the unit circle, we need to consider the signs of sin(A) and cos(A). In the given information, it is mentioned that cos(A) = \frac{\sqrt{203}}{18}, which is positive. Looking at the unit circle, we can see that cos(A) > 0 when the angle A is in the first or fourth quadrant. In these two quadrants, sin(A) is also positive. Therefore, sin(A) = \frac{\sqrt{121}}{18} = \frac{11}{18}.
Now that we have the values of sin(A) and cos(A), we can use the definition of tan(A):
tan(A) = \frac{sin(A)}{cos(A)}
= \frac{\frac{11}{18}}{\frac{\sqrt{203}}{18}}
= \frac{11}{\sqrt{203}}
To simplify the expression, we can rationalize the denominator:
tan(A) = \frac{11}{\sqrt{203}} \cdot \frac{\sqrt{203}}{\sqrt{203}}
= \frac{11\sqrt{203}}{203}
Thus, tan(A) is equal to \frac{11\sqrt{203}}{203}.