Explanation:
To find the values of the trigonometric functions, we need to know the angle whose cosine, sine, etc. we are trying to find. In this case, we are given that the cosine of the angle is 3/5, but we are not told the measure of the angle itself. We can find the values of the other trigonometric functions using the given value of the cosine, but we must first express the value in radians.
Since the cosine of an angle is equal to the x-coordinate of the point on the unit circle that is located on the terminal side of the angle, we can find the value of the angle in radians by finding the inverse cosine (cos^-1) of 3/5. In other words, we want to find the angle such that the cosine of the angle is 3/5.
The inverse cosine function is denoted by cos^-1, so we can write:
$\cos^{-1}(3/5) = \theta$
where $\theta$ is the measure of the angle in radians.
To find the value of the other trigonometric functions, we can use the following relationships:
$\cos^2\theta = 1 - \sin^2\theta$
$\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta} = \frac{1 - \cos^2\theta}{\cos^2\theta}$
$\csc^2\theta = \frac{1}{\sin^2\theta}$
$\sec^2\theta = \frac{1}{\cos^2\theta}$
$\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta} = \frac{\cos^2\theta}{1 - \cos^2\theta}$
We can now plug in the value of $\theta$ that we found earlier and use these relationships to find the values of the other trigonometric functions:
a. $\cos\theta = \cos(\cos^{-1}(3/5)) = \frac{3}{5}$
b. $\sin^2\theta = 1 - \cos^2\theta = 1 - (\frac{3}{5})^2 = \frac{4}{25}$
c. $\cos^2\theta = (\frac{3}{5})^2 = \frac{9}{25}$
d. $\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta} = \frac{\frac{4}{25}}{\frac{9}{25}} = \frac{4}{9}$
e. $\csc^2\theta = \frac{1}{\sin^2\theta} = \frac{1}{\frac{4}{25}} = \frac{25}{4}$
f. $\sec^2\theta = \frac{1}{\cos^2\theta} = \frac{1}{\frac{9}{25}} = \frac{25}{9}$
g. $\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta} = \frac{\frac{9}{25}}{\frac