Problem 24 - 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. $$(-24,-7)$$

Step-by-Step Solution

Answer

Sine: \\(\frac{-7}{25}\\), Cosine: \\(\frac{-24}{25}\\), Tangent: \\(\frac{7}{24}\\), Cosecant: \\(-\frac{25}{7}\\), Secant: \\(-\frac{25}{24}\\), Cotangent: \\(\frac{24}{7}\\).

Step 1: Plot the Point and Determine the Quadrant

The point \((-24, -7)\) is located in the third quadrant of the Cartesian plane because both coordinates are negative.

Step 2: Compute the radius (r)

The radius \(r\) can be found using the Pythagorean theorem: \[ r = \sqrt{(-24)^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25. \]

Step 3: Calculate the Sine Function

The sine of angle \(\theta\) is given by \(\sin \theta = \frac{y}{r} = \frac{-7}{25}.\)

Step 4: Calculate the Cosine Function

The cosine of angle \(\theta\) is given by \(\cos \theta = \frac{x}{r} = \frac{-24}{25}.\)

Step 5: Calculate the Tangent Function

The tangent of angle \(\theta\) is given by \(\tan \theta = \frac{y}{x} = \frac{-7}{-24} = \frac{7}{24}.\)

Step 6: Calculate the Cosecant Function

The cosecant of angle \(\theta\) is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta} = \frac{25}{-7}\). The rationalized form is \(-\frac{25}{7}\).

Step 7: Calculate the Secant Function

The secant of angle \(\theta\) is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta} = \frac{25}{-24}\). The rationalized form is \(-\frac{25}{24}\).

Step 8: Calculate the Cotangent Function

The cotangent of angle \(\theta\) is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta} = \frac{24}{7}.\)

Key Concepts

Cartesian plane

The Cartesian plane is fundamental in understanding and graphically representing mathematical relationships, especially in geometry and trigonometry. It is essentially a two-dimensional plane that is created by the intersection of two number lines: the horizontal x-axis and the vertical y-axis. These axes divide the plane into four quadrants, each representing a unique combination of positive and negative values for x and y.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Understanding which quadrant a point is in helps determine the signs of trigonometric functions. In the exercise, the point \((-24, -7)\) is located in the third quadrant, indicating both coordinates are negative.

Pythagorean theorem

The Pythagorean theorem is a key concept used to find the length of sides in a right triangle. In the context of trigonometry, it helps link the coordinates of a point to the hypotenuse (or radius) of the triangle formed when sketching an angle. The theorem states that in a right triangle:\[ c^2 = a^2 + b^2 \] where \(c\) represents the hypotenuse, and \(a\) and \(b\) represent the other two sides.

For the point \((-24, -7)\), you can use this theorem to find the radius \(r\), which serves as the hypotenuse in trigonometric terms:

• \((-24)^2 + (-7)^2 = 576 + 49 = 625\)
• \[ r = \sqrt{625} = 25 \]

This radius \(r\) is crucial in computing the values of the various trigonometric functions.

standard position

Angles in standard position give a common baseline for measuring and comparing angles. An angle is said to be in standard position when its vertex is located at the origin \((0,0)\) of the Cartesian plane, and its initial side lies along the positive x-axis. The terminal side of the angle determines its specific position and, ultimately, the location of any points associated with it.

This framework simplifies finding trigonometric function values because it provides a consistent reference. When we discuss the least positive measure of \(\theta\), we refer to the smallest angle formed by rotating from the positive x-axis to the terminal side where the point, such as \((-24, -7)\), is situated. The use of consistent reference points, like the positive x-axis, allows everyone to have a common understanding.

quadrant analysis

Quadrant analysis involves evaluating the placement of angles or points on the Cartesian plane to determine the characteristics of trigonometric functions. Each quadrant contains angles with unique properties that influence the sign of sine, cosine, and tangent functions. Understanding this can simplify solving trigonometric problems.

For instance:

• Quadrant I: Angles here have both sine and cosine as positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

In the exercise, the point \((-24, -7)\) is in the third quadrant, meaning both sine and cosine values will be negative. This directly affects the trigonometric function calculations and ultimately, their interpretations. Such analysis also helps predict the behavior of functions like tangent, which in the third quadrant becomes positive due to the double negative impact on its signs.