91Ó°ÊÓ

Inverse trigonometric functions, often called arcfunctions, are used to find the angles whose trigonometric ratios are known. The notation for the inverse sine function is \( \sin^{-1} \theta \) or \( \arcsin(\theta) \). For example, if \( \sin(\theta) = x \), then \( \sin^{-1}(x) = \theta \). These functions are extremely useful when solving for angles in right triangles, especially when you only have side lengths. In the given exercise, \( \theta = \sin^{-1} \left( \frac{p}{\sqrt{p^2 + 9}} \right) \) means we are finding the angle \( \theta \) whose sine ratio is \( \frac{p}{\sqrt{p^2 + 9}} \).

The Pythagorean Identity is a fundamental relation in trigonometry that expresses the inherent relationship between the sine and cosine of an angle. It states that \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This is derived from the Pythagorean theorem applied to the unit circle. When you are given one trigonometric function, like the sine, you can always find the cosine using this identity. In this exercise, we use it to find \( \cos(\theta) \). Since \( \sin(\theta) = \frac{p}{\sqrt{p^2 + 9}} \), we calculate \( \sin^2(\theta) \) and then substitute into the identity to find \( \cos^2(\theta) \): \( \cos^2(\theta) = 1 - \frac{p^2}{p^2 + 9} = \frac{9}{p^2 + 9} \). Finally, solving for \( \cos(\theta) \), we get \( \frac{3}{\sqrt{p^2 + 9}} \).

Trigonometric ratios are ratios of the lengths of sides in a right triangle. The three most important ratios are sine (sin), cosine (cos), and tangent (tan). For a given angle \( \theta \):

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In the exercise, we found previously \( \sin(\theta) \) and \( \cos(\theta) \). To find \( \tan(\theta) \), we use the ratio \( \frac{\sin(\theta)}{\cos(\theta)} \). Hence, substituting our values, we get \( \frac{\frac{p}{\sqrt{p^2 + 9}}}{\frac{3}{\sqrt{p^2 + 9}}} = \frac{p}{3} \). So, the tangent of angle \( \theta \) is \( \frac{p}{3} \).