Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-3,4)$$

Step-by-Step Solution

Answer

Angle \(\theta\) is 1 \( \sin(\theta) = \frac{4}{5} \), 2 \( \cos(\theta) = -\frac{3}{5} \), 3 \( \tan(\theta) = -\frac{4}{3} \), 4 \( \csc(\theta) = \frac{5}{4} \), 5 \( \sec(\theta) = -\frac{5}{3} \), 6 \( \cot(\theta) = -\frac{3}{4} \).

Step 1: Identify the Quadrant

The given point \((-3, 4)\) lies in the second quadrant because its x-coordinate is negative and the y-coordinate is positive.

Step 2: Calculate the Reference Angle

To find the reference angle, use the formula \( \tan^{-1}\left(\frac{|y|}{|x|}\right) \). Here, \(y = 4\) and \(x = -3\), so the reference angle is \( \tan^{-1}\left(\frac{4}{3}\right) \).

Step 3: Find the Angle in Standard Position

In the second quadrant, the angle in standard position is \( \theta = 180^\circ - \text{reference angle} \). Thus, \( \theta = 180^\circ - \tan^{-1}\left(\frac{4}{3}\right) \).

Step 4: Calculate the Hypotenuse Using the Pythagorean Theorem

Compute the hypotenuse \( r \) using \( r = \sqrt{x^2 + y^2} \). So, \( r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 \).

Step 5: Determine the Sine Function

The sine function is calculated by \( \sin(\theta) = \frac{y}{r} = \frac{4}{5} \).

Step 6: Determine the Cosine Function

The cosine function is \( \cos(\theta) = \frac{x}{r} = \frac{-3}{5} \).

Step 7: Determine the Tangent Function

The tangent function is calculated by \( \tan(\theta) = \frac{y}{x} = \frac{4}{-3} = -\frac{4}{3} \).

Step 8: Determine the Cosecant Function

Cosecant is the reciprocal of sine: \( \csc(\theta) = \frac{r}{y} = \frac{5}{4} \).

Step 9: Determine the Secant Function

Secant is the reciprocal of cosine: \( \sec(\theta) = \frac{r}{x} = \frac{5}{-3} = -\frac{5}{3} \).

Step 10: Determine the Cotangent Function

Cotangent is the reciprocal of tangent: \( \cot(\theta) = \frac{x}{y} = \frac{-3}{4} = -\frac{3}{4} \).

Key Concepts

Standard Position

The concept of **Standard Position** is fundamental in trigonometry because it provides a consistent way to measure angles. An angle is said to be in standard position if its vertex is at the origin of the coordinate plane and its initial side is on the positive x-axis. The terminal side is then determined by rotating the initial side, either clockwise or counterclockwise.

• Rotation Counterclockwise: Results in a positive angle.
• Rotation Clockwise: Results in a negative angle.

To understand this better, consider the angle \(\theta\) formed with point \((-3, 4)\). Since the angle must be in standard position, we start measuring from the positive x-axis. The point \((-3, 4)\) determines the terminal side of the angle in the second quadrant, reflecting the fact that x is negative and y is positive.

Reference Angle

A **Reference Angle** is an acute angle formed by the terminal side of the given angle and the x-axis. It helps in easily calculating trigonometric function values since it applies the intrinsic properties of angles in different quadrants.

• It is always between 0° and 90°.
• The formula changes based on the quadrant the angle lies in.

For an angle in the second quadrant, like in our problem where \((-3, 4)\) defines \(\theta\), the reference angle \(\alpha\) is calculated using the formula:\[\alpha = \tan^{-1}\left(\frac{|4|}{|-3|}\right) = \tan^{-1}\left(\frac{4}{3}\right)\]In this context, the reference angle aids the process of finding trigonometric function values by leveraging symmetry properties of the unit circle.

Sine, Cosine, Tangent

The three primary trigonometric functions are **Sine (sin)**, **Cosine (cos)**, and **Tangent (tan)**, which are derived from the relationships found in a right triangle. These functions allow us to relate the angles and sides of a triangle.

• **Sine** is the ratio of the length of the opposite side to the hypotenuse: \(\sin(\theta) = \frac{4}{5}\).
• **Cosine** is the ratio of the adjacent side to the hypotenuse: \(\cos(\theta) = \frac{-3}{5}\).
• **Tangent** is the ratio of the opposite side to the adjacent side: \(\tan(\theta) = -\frac{4}{3}\).

These ratios depend on the point \((-3, 4)\) which forms a right triangle with the hypotenuse calculated as 5 (using the Pythagorean theorem). These basic functions are the cornerstone for further trigonometric calculations.

Reciprocal Trigonometric Functions

The **Reciprocal Trigonometric Functions** provide additional tools for solving trigonometric problems, commonly being the reciprocals of sine, cosine, and tangent.

• **Cosecant (csc)** is the reciprocal of sine: \(\csc(\theta) = \frac{5}{4}\).
• **Secant (sec)** is the reciprocal of cosine: \(\sec(\theta) = -\frac{5}{3}\).
• **Cotangent (cot)** is the reciprocal of tangent: \(\cot(\theta) = -\frac{3}{4}\).

These functions are particularly useful in scenarios where division by zero concerns arise or when dealing with specific trigonometric identities. They provide flexibility in transforming complex trigonometric expressions into simpler forms and are essential for solving many geometric and real-world problems.