What Is The Exact Value Of Cos 195

What is the Exact Value of cos 195°? A Comprehensive Guide

Determining the exact value of trigonometric functions like cos 195° often requires leveraging trigonometric identities and understanding the unit circle. This isn't a simple matter of plugging a number into a calculator; finding the exact value demands a deeper understanding of trigonometric principles. This comprehensive guide will walk you through several methods to calculate cos 195°, explaining the reasoning behind each step and highlighting the importance of choosing the most efficient approach.

Understanding the Unit Circle and Trigonometric Identities

Before diving into the calculation, let's refresh our understanding of the unit circle and key trigonometric identities. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Several identities will be crucial in our calculations:

• Sum and Difference Identities: These are fundamental. For cosine, they are:

  • cos(A + B) = cos A cos B - sin A sin B
  • cos(A - B) = cos A cos B + sin A sin B

• Half-Angle Identities: These are incredibly useful for simplifying angles. For cosine:

  • cos(A/2) = ±√[(1 + cos A)/2] (The sign depends on the quadrant of A/2)

• Double-Angle Identities: These can be derived from the sum identities:

  • cos(2A) = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A

• Co-function Identities: These relate sine and cosine:

  • cos(90° - θ) = sin θ
  • sin(90° - θ) = cos θ

These identities provide multiple pathways to solving for cos 195°. Let's explore the most efficient methods.

Method 1: Using the Sum Identity (cos(150° + 45°))

This method breaks down 195° into the sum of two angles whose cosine and sine values are readily known: 150° and 45°.

1. Apply the Sum Identity: We use the identity cos(A + B) = cos A cos B - sin A sin B, where A = 150° and B = 45°.

2. Substitute Known Values: We know that:

   • cos 150° = -√3/2
   • sin 150° = 1/2
   • cos 45° = √2/2
   • sin 45° = √2/2

3. Calculate: cos 195° = cos(150° + 45°) = (cos 150°)(cos 45°) - (sin 150°)(sin 45°) cos 195° = (-√3/2)(√2/2) - (1/2)(√2/2) cos 195° = -√6/4 - √2/4 cos 195° = (-√6 - √2)/4

This is our exact value. This method is relatively straightforward, leveraging easily remembered values for 45° and 150°.

Method 2: Using the Difference Identity (cos(240° - 45°))

Alternatively, we can express 195° as the difference between two known angles: 240° and 45°.

1. Apply the Difference Identity: We use the identity cos(A - B) = cos A cos B + sin A sin B, where A = 240° and B = 45°.

2. Substitute Known Values: We know that:

   • cos 240° = -1/2
   • sin 240° = -√3/2
   • cos 45° = √2/2
   • sin 45° = √2/2

3. Calculate: cos 195° = cos(240° - 45°) = (cos 240°)(cos 45°) + (sin 240°)(sin 45°) cos 195° = (-1/2)(√2/2) + (-√3/2)(√2/2) cos 195° = -√2/4 - √6/4 cos 195° = (-√2 - √6)/4

This yields the same result as Method 1, confirming the accuracy of our calculations.

Method 3: Using the Half-Angle Identity

This method is slightly more involved but demonstrates another powerful technique. We can express 195° as half of 390°.

1. Apply the Half-Angle Identity: We use the identity cos(A/2) = ±√[(1 + cos A)/2], where A = 390°.

2. Determine the Sign: Since 195° lies in the third quadrant, where cosine is negative, we choose the negative square root.

3. Substitute and Calculate: cos 195° = cos(390°/2) = -√[(1 + cos 390°)/2] Since cos 390° = cos (360° + 30°) = cos 30° = √3/2, we substitute: cos 195° = -√[(1 + √3/2)/2] cos 195° = -√[(2 + √3)/4] cos 195° = -√(2 + √3)/2

This expression seems different from our previous results. However, let's rationalize it:

To show that this is equivalent, requires further manipulation. This approach, while valid, is less efficient than the sum or difference identities for this specific problem because it requires extra steps to simplify the radical expression to match the previous results. It demonstrates the power of the half-angle identity, but for cos 195°, the sum/difference approach is cleaner.

Choosing the Best Method

While all three methods are mathematically correct, the first two (using the sum and difference identities) are generally preferred for cos 195° due to their simplicity and efficiency. They directly lead to a simplified, exact value without requiring additional algebraic manipulation to rationalize the denominator or simplify complex radicals as the half-angle method might. The choice of method often depends on the specific angle and the available information. Recognizing the most efficient path is a crucial skill in trigonometry.

Applications and Further Exploration

Understanding how to find the exact value of trigonometric functions is essential in various fields, including:

• Calculus: Exact values are crucial for simplifying derivatives and integrals involving trigonometric functions.

• Physics and Engineering: Many physical phenomena are modeled using trigonometric functions, and exact values are necessary for precise calculations.

• Computer Graphics and Game Development: Accurate calculations of angles and positions heavily rely on trigonometric functions.

• Navigation and Surveying: Trigonometry plays a fundamental role in determining distances and positions.

This exploration of finding cos 195° goes beyond a simple numerical answer; it emphasizes the importance of mastering trigonometric identities and selecting the most effective approach for efficient problem-solving. The ability to manipulate trigonometric expressions is a cornerstone of mathematical proficiency across various disciplines. Further exploration could involve investigating similar calculations for other angles, exploring the relationship between different trigonometric identities, or delving into more advanced trigonometric concepts. Remember, practice is key to mastering these skills.