Question

In Exercises \(37-40\) , use the given information to find the values of the six trigonometric functions at the angle \(\theta\) . Give exact answers. The point \(P(-3,4)\) is on the terminal side of \(\theta\)

Short answer

The exact values of the six trigonometric functions for the angle \(\theta\) are: \(\sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{-3}{5}, \tan(\theta) = -\frac{4}{3}, \csc(\theta) = \frac{5}{4}, \sec(\theta) = -\frac{5}{3}, \cot(\theta) = -\frac{3}{4}\)

Step-by-step solution

Calculate the Hypotenuse of the triangle formed by the Coordinates of Point P

We are given the point \(P(-3, 4)\). Let's denote \(x = -3\) and \(y = 4\) for simplicity. The hypotenuse \(r\) of the triangle is given by the Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\). Substituting the values, we get \(r = \sqrt{(-3)^2+(4)^2} = 5\).

Evaluate Trigonometric Functions

We can express the six trigonometric functions in terms of \(x\), \(y\), and \(r\). Here are their respective definitions: \(\sin(\theta) = \frac{y}{r}, \cos(\theta) = \frac{x}{r}, \tan(\theta) = \frac{y}{x}, \csc(\theta) = \frac{r}{y}, \sec(\theta) = \frac{r}{x}, \cot(\theta) = \frac{x}{y}\). Substituting the values, we obtain: \(\sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{-3}{5}, \tan(\theta) = -\frac{4}{3}, \csc(\theta) = \frac{5}{4}, \sec(\theta) = -\frac{5}{3}, \cot(\theta) = -\frac{3}{4}\)

Interpret Results

The signs in our results denote the quadrant in which the terminal side of the angle \(\theta\) lies. Here, both sine and cosine are positive which means the angle \(\theta\) is in the second quadrant, indicating our understanding aligns with the given point P(-3,4).