Understanding trigonometric identities and equations is crucial in advanced mathematics. In this article, we will explore the value of sec θ given the equation sec θ - 2 cos θ 7/2, with θ being a positive acute angle. We will solve the equation step by step, and discuss the significance of the solution.

Introduction to Trigonometric Identities

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. Trigonometric identities are fundamental equations that represent these relationships, and they are widely used in both theoretical and applied mathematics.

Understanding the Given Equation

The equation provided for our solution is:

sec θ - 2 cos θ 7/2

Here, sec θ is the secant of the angle θ, a reciprocal function of the cosine function.

Solving the Equation

Let's start by rewriting sec θ as a fraction:

sec θ p

Substituting sec θ with p, we have:

p - 2 cos θ 7/2

Now, we need to express cos θ in terms of p:

cos θ 1/p

Substituting this into the equation:

p - 2(1/p) 7/2

Multiplying through by p to clear the fraction:

p^2 - 2 7/2 p

Multiplying the entire equation by 2 to clear the fraction:

2p^2 - 4 7p

Shifting all terms to one side to form a quadratic equation:

2p^2 - 7p - 4 0

This is a standard quadratic equation of the form:

p^2 - 3.5p - 2 0

Solving the Quadratic Equation

The quadratic equation can be solved using the quadratic formula:

p (-b ± √(b^2 - 4ac)) / (2a)

Here, a 2, b -7, and c -4.

Substituting these values into the quadratic formula:

p (7 ± √((-7)^2 - 4(2)(-4))) / (2(2))

Simplifying the expression inside the square root:

p (7 ± √(49 32)) / 4

p (7 ± √81) / 4

p (7 ± 9) / 4

This gives us two possible solutions for p:

p (7 9) / 4 16 / 4 4

p (7 - 9) / 4 -2 / 4 -0.5

Since sec θ is positive for a positive acute angle, we discard the negative solution:

Therefore, p 4

Given that sec θ p, sec θ 4 is the final value.

Conclusion: By solving the equation sec θ - 2 cos θ 7/2, we determined that sec θ 4 for a positive acute angle θ. Understanding such equations is crucial for advanced mathematical problem-solving in fields such as physics, engineering, and data science.