Find The Six Trigonometric Function Values Of The Specified Angle

Finding the Six Trigonometric Function Values of a Specified Angle

Determining the six trigonometric function values (sine, cosine, tangent, cosecant, secant, and cotangent) for a given angle is a fundamental concept in trigonometry. This process involves understanding the unit circle, reference angles, and the signs of trigonometric functions in different quadrants. This comprehensive guide will walk you through the process step-by-step, providing examples and clarifying common pitfalls.

Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane. It's a crucial tool for visualizing trigonometric functions. Any point (x, y) on the unit circle can be represented by the coordinates (cos θ, sin θ), where θ is the angle formed by the positive x-axis and the line segment connecting the origin to the point.

Key takeaway: The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.

Visualizing Trigonometric Functions on the Unit Circle

Imagine a point P on the unit circle corresponding to an angle θ. The coordinates of P are (cos θ, sin θ). We can then define the other trigonometric functions:

• Tangent (tan θ): tan θ = sin θ / cos θ = y / x
• Secant (sec θ): sec θ = 1 / cos θ = 1 / x
• Cosecant (csc θ): csc θ = 1 / sin θ = 1 / y
• Cotangent (cot θ): cot θ = cos θ / sin θ = x / y

Important Note: These functions are undefined when their denominators are zero. For example, tan θ is undefined when cos θ = 0 (at θ = 90° and 270°), and sec θ is undefined when cos θ = 0. Similarly, cot θ is undefined when sin θ = 0 (at θ = 0° and 180°), and csc θ is undefined when sin θ = 0.

Determining Trigonometric Function Values: Step-by-Step Guide

Let's break down the process of finding the six trigonometric function values for a given angle. We'll consider different scenarios, including angles in standard position and angles outside the range of 0° to 360°.

Step 1: Determine the Quadrant

The first step is to identify the quadrant in which the angle lies. This determines the signs of the trigonometric functions.

• Quadrant I (0° to 90°): All trigonometric functions are positive.
• Quadrant II (90° to 180°): Only sine and cosecant are positive.
• Quadrant III (180° to 270°): Only tangent and cotangent are positive.
• Quadrant IV (270° to 360°): Only cosine and secant are positive.

This can be remembered using the mnemonic "All Students Take Calculus," where A, S, T, and C represent the positive functions in each quadrant.

Step 2: Find the Reference Angle

The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. It's always a positive angle less than 90°. To find the reference angle:

• Quadrant I: The reference angle is the angle itself.
• Quadrant II: The reference angle is 180° - θ.
• Quadrant III: The reference angle is θ - 180°.
• Quadrant IV: The reference angle is 360° - θ.

Step 3: Determine the Trigonometric Function Values of the Reference Angle

Using the reference angle, you can find the trigonometric function values using known values (like those for 30°, 45°, and 60°) or a calculator. Remember to use the unit circle to visualize these values.

Step 4: Determine the Signs of the Trigonometric Functions

Based on the quadrant identified in Step 1, assign the appropriate signs to the trigonometric function values obtained in Step 3.

Step 5: Write Out the Six Trigonometric Function Values

Examples: Finding Trigonometric Function Values

Let's work through some examples to solidify our understanding.

Example 1: Finding the trigonometric function values of 150°

1. Quadrant: 150° lies in Quadrant II.

2. Reference Angle: 180° - 150° = 30°

3. Trigonometric Function Values of the Reference Angle (30°):

   • sin 30° = 1/2
   • cos 30° = √3/2
   • tan 30° = 1/√3
   • csc 30° = 2
   • sec 30° = 2/√3
   • cot 30° = √3

4. Signs (Quadrant II): sin and csc are positive; others are negative.

5. Trigonometric Function Values of 150°:

   • sin 150° = 1/2
   • cos 150° = -√3/2
   • tan 150° = -1/√3
   • csc 150° = 2
   • sec 150° = -2/√3
   • cot 150° = -√3

Example 2: Finding the trigonometric function values of 225°

1. Quadrant: 225° lies in Quadrant III.

2. Reference Angle: 225° - 180° = 45°

3. Trigonometric Function Values of the Reference Angle (45°):

   • sin 45° = √2/2
   • cos 45° = √2/2
   • tan 45° = 1
   • csc 45° = √2
   • sec 45° = √2
   • cot 45° = 1

4. Signs (Quadrant III): tan and cot are positive; others are negative.

5. Trigonometric Function Values of 225°:

   • sin 225° = -√2/2
   • cos 225° = -√2/2
   • tan 225° = 1
   • csc 225° = -√2
   • sec 225° = -√2
   • cot 225° = 1

Example 3: Angles Greater Than 360° or Negative Angles

For angles greater than 360°, subtract multiples of 360° until you get an angle between 0° and 360°. For negative angles, add multiples of 360° until you get an angle between 0° and 360°. Then follow the steps above.

Using a Calculator

While the process above is crucial for understanding the underlying principles, you can also use a calculator to find trigonometric function values. Make sure your calculator is in the correct angle mode (degrees or radians).

Applications of Trigonometric Function Values

Finding trigonometric function values is fundamental to numerous applications in various fields:

• Physics: Analyzing projectile motion, forces, and wave phenomena.
• Engineering: Solving problems related to structures, mechanics, and surveying.
• Navigation: Determining distances and directions using triangulation.
• Computer Graphics: Generating realistic images and animations.
• Signal Processing: Analyzing and manipulating signals.

Conclusion

Mastering the ability to find the six trigonometric function values for any given angle is essential for anyone studying trigonometry and its applications. By understanding the unit circle, reference angles, and the signs of functions in different quadrants, you can confidently tackle these calculations and apply them to a wide range of problems. Remember to practice regularly to build your proficiency and deepen your understanding. The examples provided offer a solid foundation for tackling more complex scenarios and solidifying your understanding of this crucial trigonometric concept. Remember to always double-check your work and use appropriate tools, such as calculators, to ensure accuracy.