Let (8, -6) Be A Point On The Terminal Side Of An Angle \(\theta\) In Standard Position. Find The Exact Value Of
---
Introduction
Understanding how to find the exact value of trigonometric functions for a given point on the terminal side of an angle in standard position is fundamental in mathematics, especially in trigonometry. When a point \((x, y)\) lies on the terminal side of an angle \(\theta\) in the coordinate plane, the goal is often to determine the exact values of the sine, cosine, tangent, and other related functions associated with \(\theta\). In this article, we will explore the process of finding these values step-by-step, using the specific point \((8, -6)\) as our example.
---
The Concept of Standard Position and Coordinates
What Is an Angle in Standard Position?
An angle \(\theta\) is said to be in standard position when:
- Its vertex is at the origin \((0, 0)\).
- Its initial side lies along the positive x-axis.
- Its terminal side is determined by the ray passing through a point \((x, y)\) in the coordinate plane.
Significance of the Point \((8, -6)\)
- The point \((8, -6)\) lies somewhere in the fourth quadrant because:
- \(x = 8 > 0\)
- \(y = -6 < 0\)
- The coordinates give us the position of the terminal side of the angle \(\theta\) in the coordinate system.
Visual Representation
Visualizing this point helps in understanding the angle's position:
- Draw the coordinate axes.
- Plot the point \((8, -6)\).
- Draw a line segment from the origin to this point.
- The angle \(\theta\) is the measure between the positive x-axis and this line segment.
---
Step-by-Step Process to Find Exact Trigonometric Values
Step 1: Calculate the Distance from the Origin to the Point
This distance, called the radius or hypotenuse \(r\), is essential for calculating trigonometric functions.
\[
r = \sqrt{x^2 + y^2}
\]
Applying the given coordinates:
\[
r = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]
Step 2: Determine the Basic Trigonometric Functions
Using the coordinates and the hypotenuse, the basic functions are:
- Sine \(\sin \theta\):
\[
\sin \theta = \frac{y}{r} = \frac{-6}{10} = -\frac{3}{5}
\]
- Cosine \(\cos \theta\):
\[
\cos \theta = \frac{x}{r} = \frac{8}{10} = \frac{4}{5}
\]
- Tangent \(\tan \theta\):
\[
\tan \theta = \frac{y}{x} = \frac{-6}{8} = -\frac{3}{4}
\]
Step 3: Find the Reciprocal Functions
- Cosecant \(\csc \theta\):
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}
\]
- Secant \(\sec \theta\):
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}
\]
- Cotangent \(\cot \theta\):
\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}
\]
Step 4: Summarize the Exact Values
| Function | Exact Value | Explanation |
|------------|--------------------------------|----------------------------------------------|
| \(\sin \theta\) | \(-\frac{3}{5}\) | \(y/r = -6/10\) |
| \(\cos \theta\) | \(\frac{4}{5}\) | \(x/r = 8/10\) |
| \(\tan \theta\) | \(-\frac{3}{4}\) | \(y/x = -6/8\) |
| \(\csc \theta\) | \(-\frac{5}{3}\) | reciprocal of \(\sin \theta\) |
| \(\sec \theta\) | \(\frac{5}{4}\) | reciprocal of \(\cos \theta\) |
| \(\cot \theta\) | \(-\frac{4}{3}\) | reciprocal of \(\tan \theta\) |
---
Understanding the Quadrant and Sign Conventions
Quadrant Location
Since the point \((8, -6)\) lies in the fourth quadrant:
- \(\sin \theta\) is negative.
- \(\cos \theta\) is positive.
- \(\tan \theta\) is negative.
These signs are consistent with the computed values.
Significance for Trigonometric Functions
Knowing the quadrant helps to determine the signs of the trigonometric functions without ambiguity.
---
Additional Insights and Applications
Why Is Finding Exact Values Important?
- Precise calculations in engineering, physics, and mathematics.
- Simplification of expressions involving trigonometric functions.
- Solving equations where approximate decimal values are insufficient.
Common Applications
- Calculating angles in physics problems involving vectors.
- Analyzing periodic functions in signal processing.
- Solving geometry problems involving triangles and circles.
---
Practice Problems for Mastery
To reinforce understanding, try solving these similar problems:
- Find the exact values of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) for the point \((-6, 8)\).
- Given the point \((3, 4)\), determine the values of all six trigonometric functions.
- For a point \((x, y)\) in the second quadrant, with \(x = -5\) and \(y = 12\), compute the exact trigonometric functions of the corresponding angle \(\theta\).
---
Conclusion
Determining the exact values of trigonometric functions for a point on the terminal side of an angle in standard position is a critical skill in mathematics. Using the point \((8, -6)\), we demonstrated how to calculate the hypotenuse, then derive the sine, cosine, tangent, and their reciprocal functions. Remember, understanding the signs based on the quadrant enhances your ability to interpret these values correctly. Mastery of these concepts enables you to solve complex problems in geometry, physics, and engineering with confidence.
---
Frequently Asked Questions (FAQs)
Q1: Why do we calculate the hypotenuse \(r\) when finding trig functions?
A: The hypotenuse \(r\) represents the distance from the origin to the point, which normalizes the coordinates and allows us to compute ratios that define the trig functions.
Q2: How do I determine the signs of the trig functions in different quadrants?
A: Use the ASTC (All Students Take Calculus) mnemonic or the signs of \(x\) and \(y\) in each quadrant:
- Quadrant I: All positive
- Quadrant II: Sine positive
- Quadrant III: Tangent positive
- Quadrant IV: Cosine positive
Q3: Can these methods be used for points outside the coordinate axes?
A: Yes, these methods are applicable for any point (except at the origin), regardless of the quadrant or location.
---
By mastering the process of calculating exact trigonometric values from coordinate points, you enhance your problem-solving toolkit for a wide range of mathematical applications.
