Solving for the Value of Cosec Theta Given sec Theta tan Theta p

In this article, we will explore how to derive and simplify the value of cosec θ when given the expression sec θ tan θ p, where p is a constant. This problem involves trigonometric identities and algebraic manipulation, offering a rich learning experience in advanced trigonometry.

Introduction

The problem at hand is to find the value of cosec θ given the equation sec θ tan θ p. Trigonometric functions play a vital role in mathematics and various fields of engineering and physics. By understanding how to manipulate and solve these equations, we can enhance our problem-solving skills and deepen our knowledge of trigonometry.

Solving the Equation

Step 1: Express sec θ and tan θ in Terms of Sine and Cosine

First, let's express the given trigonometric functions in terms of sine and cosine:

sec θ 1 / cos θ tan θ sin θ / cos θ

When we combine these, the equation becomes:

1 / cos θ × (sin θ / cos θ) p

Step 2: Simplify the Expression

Now, we can simplify the left-hand side of the equation:

(1 sin θ) / cos θ p

From this, we can isolate cos θ:

cos θ (1 sin θ) / p

Step 3: Form a Quadratic Equation

Next, we square both sides to eliminate the square root:

1 sin^2 θ p^2 (1 sin θ)^2 / p^2

Expanding the right-hand side gives:

1 sin^2 θ p^2 (1 2sin θ sin^2 θ) / p^2

Simplifying further:

1 sin^2 θ 1 2sin θ sin^2 θ

Subtracting sin^2 θ from both sides simplifies the equation:

1 2sin θ p^2 sin^2 θ - p^2

Combine like terms to form a quadratic equation:

2sin^2 θ - 2sin θ 1 - p^2 0

Step 4: Solve the Quadratic Equation

Using the quadratic formula, we find sin θ:

sin θ (-b ± √(b^2 - 4ac)) / 2a

Here, a 2, b -2, and c 1 - p^2:

sin θ (2 ± √((-2)^2 - 4 × 2 × (1 - p^2))) / 4

Simplifying the discriminant:

sin θ (2 ± √(4 - 8(1 - p^2))) / 4

sin θ (2 ± √(4 - 8 8p^2)) / 4

sin θ (2 ± 2√(p^2 - 1)) / 4

sin θ 1 ± √(p^2 - 1) / 2

Step 5: Determine the Value of Cosec θ

Finally, we can find cosec θ:

cosec θ 1 / sin θ

Substitute the values of sin θ:

cosec θ 2 / (1 ± √(p^2 - 1))

This gives us a general form for cosec θ based on the value of p. Depending on the specific value of p, further simplification may be required.

Conclusion

In summary, the value of cosec θ is derived from the given equation sec θ tan θ p. Through the use of trigonometric identities and algebraic manipulation, we have found that:

cosec θ 2 / (1 ± √(p^2 - 1))

This solution not only provides a clear step-by-step approach but also highlights the importance of understanding and applying trigonometric identities effectively.

Additional Resources

For those interested in learning more about advanced trigonometry, the following resources may be helpful:

Trigonometric Identities - This resource includes detailed explanations of various trigonometric identities. Trigonometric Equations - Learn about solving trigonometric equations and inequalities. Interactive Trigonometric Identities - Practice solving trigonometric identities interactively.

Key Points

Understanding the relationship between sec θ, tan θ, and cosec θ is crucial. The quadratic formula is a powerful tool in solving trigonometric equations. Trigonometric identities play a fundamental role in simplifying problems.

FAQs

Q: What is the difference between sec θ and cosec θ?

Sec θ is the reciprocal of cos θ, while cosec θ is the reciprocal of sin θ.

Q: How do you derive the value of sin θ from sec θ and tan θ?

By expressing sec θ and tan θ in terms of sine and cosine and using the Pythagorean identity, you can derive the value of sin θ.

Q: What is the significance of the quadratic formula in solving trigonometric equations?

The quadratic formula is crucial in solving quadratic equations, which often arise in trigonometric problems. It helps find the roots of the equation, which are essential in determining the values of trigonometric functions.