TR-34: Using Pythagorean Identities (Trigonometry series by Dennis F. Davis)

Introduction

In this tutorial, we will explore the use of Pythagorean identities in trigonometry to determine the values of all trigonometric functions from the value of just one. This guide is particularly useful for students preparing for exams like the International A Level or Edexcel, as it provides a clear framework for applying these identities effectively.

Step 1: Understand Pythagorean Identities

Pythagorean identities are fundamental relationships in trigonometry that relate the square of sine, cosine, and tangent functions. The primary identities you should know are:

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

Practical Tip: Memorize these identities as they serve as the foundation for solving various trigonometric problems.

Step 2: Identify the Known Value

Start with the value of one trigonometric function. For example, if you know the value of sin(θ):

• Determine the angle θ for which you have this value.
• Identify whether the angle is in the first, second, third, or fourth quadrant, as this affects the signs of the other functions.

Common Pitfall: Failing to consider the quadrant can lead to incorrect signs for cosine, tangent, and other functions.

Step 3: Use the Pythagorean Identity

Utilize the appropriate Pythagorean identity to find the values of the other trigonometric functions.

Example Process Using sin(θ)

1. If sin(θ) = 0.6, substitute into the identity:

   • sin²(θ) + cos²(θ) = 1
   • (0.6)² + cos²(θ) = 1
   • 0.36 + cos²(θ) = 1
   • cos²(θ) = 1 - 0.36
   • cos²(θ) = 0.64
   • cos(θ) = ±√0.64 = ±0.8

2. Determine the signs based on the quadrant:

   • If θ is in the first quadrant, cos(θ) = 0.8.
   • If θ is in the second quadrant, cos(θ) = -0.8.

Step 4: Calculate Other Functions

Once you have sine and cosine, you can find the remaining trigonometric functions.

1. Calculate tangent:

   • tan(θ) = sin(θ) / cos(θ)
   • Using our example, if sin(θ) = 0.6 and cos(θ) = 0.8:
     • tan(θ) = 0.6 / 0.8 = 0.75

2. Calculate cosecant, secant, and cotangent:

   • csc(θ) = 1/sin(θ)
   • sec(θ) = 1/cos(θ)
   • cot(θ) = 1/tan(θ)

Example Calculations:

• csc(θ) = 1/0.6 = 1.6667
• sec(θ) = 1/0.8 = 1.25
• cot(θ) = 1/0.75 = 1.3333

Conclusion

By mastering Pythagorean identities and following these steps, you can efficiently determine all trigonometric function values from a single known value. Practice with different values and quadrants to strengthen your understanding. For further practice, refer to the extra problems and drills linked in the video description. Happy studying!