Problem 71 - Precalculus: Functions and Graphs

Question

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and \(P\) is on the terminal side. $$P(4,-3)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Coordinates and Configuration

The point given is \(P(4, -3)\). This point is located in the Cartesian plane, specifically in the fourth quadrant, where the x-coordinate is positive and the y-coordinate is negative.

Step 2: Calculate the Radius (r)

The radius \(r\) or the hypotenuse of the right triangle formed can be found using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

Step 3: Compute the Sine Function

Sine is defined as the ratio of the opposite side to the hypotenuse. For the angle \(\theta\), \[ \sin(\theta) = \frac{y}{r} = \frac{-3}{5}. \]

Step 4: Compute the Cosine Function

Cosine is the ratio of the adjacent side to the hypotenuse. Therefore, \[ \cos(\theta) = \frac{x}{r} = \frac{4}{5}. \]

Step 5: Compute the Tangent Function

Tangent is the ratio of the opposite side to the adjacent side. Thus, \[ \tan(\theta) = \frac{y}{x} = \frac{-3}{4}. \]

Step 6: Compute the Cosecant Function

Cosecant is the reciprocal of sine. Hence, \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{-3} = -\frac{5}{3}. \]

Step 7: Compute the Secant Function

Secant is the reciprocal of cosine. Therefore, \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4}. \]

Step 8: Compute the Cotangent Function

Cotangent is the reciprocal of tangent. Thus, \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{-3} = -\frac{4}{3}. \]

Key Concepts

Understanding Standard Position in Trigonometry

When we talk about angles in trigonometry, particularly concerning their position, the concept of 'standard position' is essential. An angle is said to be in standard position if its vertex is placed at the origin of a Cartesian plane, and its initial side lies along the positive x-axis. This setup allows us to describe angles systematically, which is especially important when defining and calculating trigonometric functions.
The terminal side of the angle will rotate from the initial side. This side determines the coordinates from which we derive many important trigonometric concepts. For example, if you are given a point on the terminal side, this point can be used to find the trigonometric functions based on its coordinates.

• The initial side is always fixed along the positive x-axis.
• The terminal side is determined by rotating either counterclockwise (positive angle) or clockwise (negative angle).
• This consistent positioning is vital for accurately describing angles in trigonometry.

By understanding standard position, one can better grasp how angles interact with the x and y coordinates, forming one of the foundations for solving trigonometric exercises.

Exploring the Cartesian Plane

The Cartesian plane, a fundamental concept in mathematics, is a two-dimensional surface defined by an x-axis (horizontal) and a y-axis (vertical). Together, these axes divide the plane into four quadrants. Each point on the plane can be described using an ordered pair (x, y).
This system is named after René Descartes, who laid the groundwork for combining algebra and geometry through this method of plotting points. It allows for the visual representation of equations and their solutions.

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

Understanding which quadrant a point lies in helps determine the sign of trigonometric ratios. For example, in the given problem, the point (4, -3) lies in the fourth quadrant, where sine is negative and cosine is positive. This information is crucial when calculating the trigonometric functions for angles in different quadrants.

The Role of the Pythagorean Theorem in Trigonometry

The Pythagorean theorem is a pivotal element in trigonometry for relating the sides of a right-angled triangle. For a triangle with legs of lengths 'a' and 'b', and hypotenuse 'c', the theorem states \(a^2 + b^2 = c^2\).
In trigonometry, the hypotenuse is often referred to as the 'radius' when working with angles in standard position on the Cartesian plane. The theorem becomes instrumental when calculating the radius, as shown in the exercise using a point's coordinates.

• The theorem enables us to find unknown side lengths using the relationship between known sides.
• It helps transform Cartesian coordinates into polar form.
• A common use is to find the hypotenuse for trigonometric functions: sine, cosine, and tangent.

For example, with point \( P(4, -3) \), the Pythagorean theorem allows us to calculate the hypotenuse \( r = 5 \). This 'r' serves as the denominator in trigonometric ratios, ensuring that we measure angles and distances accurately when solving trigonometric problems.