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Chapter 4: Problem 1

In Exercises \(1-8,\) a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta\). $$(-4,3)$$

Short Answer

Expert verified
The six trigonometric functions for the given terminal point \((-4,3)\) are: \(\sin(\theta) = \frac{3}{5}, \cos(\theta) = \frac{-4}{5}, \tan(\theta) = -\frac{3}{4}, \csc(\theta) = \frac{5}{3}, \sec(\theta) = \frac{-5}{4}, \cot(\theta) = \frac{-4}{3}\)

Step by step solution

01

Identify the terminal point coordinates

The coordinates of the terminal point given are \(-4,3\). This means that the x-coordinate (adjacent side in a right triangle with respect to angle \(\theta\)) is \(-4\) and y-coordinate (opposite side) is \(3\).
02

Find the Hypotenuse/Radii

The hypotenuse, also called the radii in this context, is determined by the Pythagorean theorem. We use the absolute values because distances are always positive. So, \(r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16+9} = \sqrt{25}=5\)
03

Determine the Six Trigonometric Functions

The six trigonometric functions are: \begin{itemize} \item Sine: \(\sin(\theta) = \frac{y}{r} = \frac{3}{5}\) \item Cosine : \(\cos(\theta) = \frac{x}{r} = \frac{-4}{5}\) \item Tangent: \(\tan(\theta) = \frac{y}{x} = \frac{3}{-4} = -\frac{3}{4}\) \item Cosecant : reciprocal of sine : \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3}\) \item Secant : reciprocal of cosine : \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{-5}{4}\) \item Cotangent : reciprocal of tangent : \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{-4}{3}\) \end{itemize}

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