Problem 36 - Precalculus: A Concise Course

Question

The terminal side of \(\boldsymbol{\theta}\) lies on the given line in the specified quadrant. Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) by finding a point on the line. $$ 4 x+3 y=0 $$ , IV

Step-by-Step Solution

Answer

The six trigonometric functions are \(\sin{\theta}=-\frac{4}{5}\), \(\cos{\theta}=\frac{3}{5}\), \(\tan{\theta}=-\frac{4}{3}\), \(\csc{\theta}=-\frac{5}{4}\), \(\sec{\theta}=\frac{5}{3}\), and \(\cot{\theta}=-\frac{3}{4}\).

Step 1: Find a Point on the Line

Since the line equation is given by \(4x + 3y = 0\), one way to find a point on this line is to set one of the variables, say x, to 1 (since we're in the IV Quadrant, x has to be positive and y negative). So we get the point \((1,-4/3)\).

Step 2: Calculate 'r'

The distance 'r' from the origin is given by \(r = \sqrt{x^2+y^2}\). Using the coordinates of our point (1, -4/3), we find that \(r = \sqrt{(1)^2 + (-\frac{4}{3})^2} = \sqrt{\frac{25}{9}} = \frac{5}{3}\).

Step 3: Calculate Six Trigonometric Functions

In the fourth quadrant, sine and its reciprocal function, cosecant, are negative, while the others are positive. So we can find the values of the trigonometric functions as follows:\(\sin{\theta} = \frac{y}{r} = -\frac{4}{5}\), \(\cos{\theta} = \frac{x}{r} = \frac{3}{5}\), \(\tan{\theta} = \frac{y}{x} = -\frac{4}{3}\), \(\csc{\theta} = \frac{1}{\sin{\theta}} = -\frac{5}{4}\), \(\sec{\theta} = \frac{1}{\cos{\theta}} = \frac{5}{3}\), \(\cot{\theta} = \frac{1}{\tan{\theta}} = -\frac{3}{4}\).

Key Concepts

Terminal Side

## Terminal SideThe concept of the terminal side is crucial in understanding angles and their trigonometric properties. When you have an angle in standard position, its initial side is on the positive x-axis, and the terminal side is where the angle ends.
The position of this terminal side determines many of the angle's properties, including the quadrant in which it lies.

• If the terminal side is in the IV Quadrant, it means the angle is between 270 and 360 degrees or \( \frac{3\pi}{2} \, \text{to} \, 2\pi \) radians.
• This positioning directly influences the signs of the trigonometric ratios calculated from that angle.

Understanding where the terminal side lies is essential for determining the correct signs of trigonometric functions and thus solving problems accurately.

IV Quadrant

## IV Quadrant The fourth quadrant of the Cartesian coordinate system is a unique area where certain trigonometric functions take on different characteristics. Angles whose terminal sides fall in this quadrant have specific properties.

• In the IV Quadrant, the x-values are positive and the y-values are negative.
• This affects the signs of trigonometric functions related to those coordinates:

- **Positive Functions**: Cosine and secant are positive in this quadrant because they involve the x-coordinate. - **Negative Functions**: Sine, cosecant, tangents, and cotangent are negative because they involve the y-coordinate or ratios that result in a negative value. Recognizing these sign patterns is crucial for solving problems involving trigonometric functions in the IV Quadrant.

Pythagorean Theorem

## Pythagorean TheoremThe Pythagorean Theorem is a foundational principle in geometry and is invaluable when working with trigonometric functions. It states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.For a given point \( (x, y) \) on the terminal side of an angle \( \theta \), the theorem helps us find \( r \), the distance from the origin to that point (hypotenuse):\[ r = \sqrt{x^2 + y^2} \]In our exercise, with the point \( (1, \, -\frac{4}{3}) \), application of the Pythagorean Theorem finds \( r = \frac{5}{3} \). Here, \( r \) is crucial for determining the trigonometric ratios by serving as the denominator in our sine, cosine, and other trigonometric functions.

Trigonometric Ratios

## Trigonometric RatiosTrigonometric ratios compare the lengths of the sides of right triangles and are fundamental in trigonometry. These ratios are essential for defining the trigonometric functions sine, cosine, tangent, and their reciprocals. The key ratios are defined as:

• **Sine** (\( \sin \theta \)): The ratio of the opposite side to the hypotenuse (\( y/r \))
• **Cosine** (\( \cos \theta \)): The ratio of the adjacent side to the hypotenuse (\( x/r \))
• **Tangent** (\( \tan \theta \)): The ratio of the opposite side to the adjacent side (\( y/x \))
• The reciprocals add more dimension with **Cosecant** (\( \csc \theta = 1/\sin \theta \)), **Secant** (\( \sec \theta = 1/\cos \theta \)), and **Cotangent** (\( \cot \theta = 1/\tan \theta \)).

For a point in the IV Quadrant, remembering the sign changes (- for sine and related functions; + for cosine-related) is crucial. Practicing these concepts solidifies understanding and improves problem-solving skills.