Understanding the Secant Function

Oct 13, 2025 by windailys.com 55 views

Introduction

Trigonometric equations are a fundamental aspect of mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will focus on solving a specific trigonometric equation involving the secant function. We will determine the solution set of $\theta$ where $90^{\circ} \leq \theta \leq 360^{\circ}$ for $\sec(\theta + 50^{\circ}) = 2$ .

Understanding the Secant Function

The secant function is the reciprocal of the cosine function. It is defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$ . The secant function has a period of $360^{\circ}$ and is positive in the first and fourth quadrants.

The Given Equation

The given equation is $\sec(\theta + 50^{\circ}) = 2$ . To solve this equation, we need to find the values of $\theta$ that satisfy the equation.

Step 1: Rewrite the Equation

We can rewrite the equation as $\frac{1}{\cos(\theta + 50^{\circ})} = 2$ . This is equivalent to $\cos(\theta + 50^{\circ}) = \frac{1}{2}$ .

Step 2: Find the Values of $\theta$

To find the values of $\theta$ , we need to find the angles that satisfy the equation $\cos(\theta + 50^{\circ}) = \frac{1}{2}$ . We know that the cosine function has a period of $360^{\circ}$ , so we can restrict our search to the interval $90^{\circ} \leq \theta \leq 360^{\circ}$ .

Using the Unit Circle

The unit circle is a powerful tool for solving trigonometric equations. We can use the unit circle to find the values of $\theta$ that satisfy the equation.

Step 3: Find the Reference Angle

The reference angle is the acute angle between the terminal side of the angle and the x-axis. We can find the reference angle by subtracting $50^{\circ}$ from $\theta$ .

Step 4: Find the Values of $\theta$

We can use the reference angle to find the values of $\theta$ that satisfy the equation. We know that the cosine function is positive in the first and fourth quadrants, so we can restrict our search to these quadrants.

Solving for $\theta$

We can solve for $\theta$ by using the inverse cosine function. We have:

\theta + 50^{\circ} = \cos^{-1}\left(\frac{1}{2}\right)

\theta = \cos^{-1}\left(\frac{1}{2}\right) - 50^{\circ}

Evaluating the Inverse Cosine Function

We can evaluate the inverse cosine function using a calculator or a trigonometric table. We have:

\cos^{-1}\left(\frac{1}{2}\right) = 60^{\circ}

Substituting the Value

We can substitute the value of the inverse cosine function into the equation for $\theta$ . We have:

\theta = 60^{\circ} - 50^{\circ}

\theta = 10^{\circ}

Checking the Solution

We can check the solution by plugging it back into the original equation. We have:

\sec(10^{\circ} + 50^{\circ}) = \sec(60^{\circ})

\sec(60^{\circ}) = 2

Conclusion

We have solved the trigonometric equation $\sec(\theta + 50^{\circ}) = 2$ and found the solution set of $\theta$ where $90^{\circ} \leq \theta \leq 360^{\circ}$ . The solution is $\theta = 10^{\circ}$ .

Additional Solutions

We can find additional solutions by using the periodicity of the secant function. We have:

\sec(\theta + 360^{\circ}) = \sec(\theta)

We can use this property to find additional solutions.

Step 5: Find Additional Solutions

We can find additional solutions by adding $360^{\circ}$ to the original solution. We have:

\theta = 10^{\circ} + 360^{\circ}

\theta = 370^{\circ}

Checking the Additional Solution

We can check the additional solution by plugging it back into the original equation. We have:

\sec(370^{\circ} + 50^{\circ}) = \sec(420^{\circ})

\sec(420^{\circ}) = 2

Conclusion

We have found additional solutions to the trigonometric equation $\sec(\theta + 50^{\circ}) = 2$ . The additional solutions are $\theta = 370^{\circ}$ .

Final Solution Set

The final solution set is $\theta = 10^{\circ}, 370^{\circ}$ .

Conclusion

$In this article, we have solved the trigonometric equation $\sec(\theta + 50^{\circ}) = 2$ and found the solution set of $\theta$ where $90^{\circ} \leq \theta \leq 360^{\circ}$. The solution set is $\theta = 10^{\circ}, 370^{\circ}$. We have used the unit circle and the periodicity of the secant function to find the solutions.<br/> **Solving Trigonometric Equations: A Comprehensive Approach**$

Q&A: Solving Trigonometric Equations

Q: What is the secant function?

A: The secant function is the reciprocal of the cosine function. It is defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$ .

Q: What is the period of the secant function?

A: The period of the secant function is $360^{\circ}$ .

Q: What is the range of the secant function?

A: The range of the secant function is all real numbers.

Q: How do you solve a trigonometric equation involving the secant function?

A: To solve a trigonometric equation involving the secant function, you need to use the reciprocal identity of the secant function, which is $\sec(\theta) = \frac{1}{\cos(\theta)}$ . You can then use the properties of the cosine function to solve the equation.

Q: What is the difference between the secant and cosine functions?

A: The secant function is the reciprocal of the cosine function. This means that the secant function is equal to 1 divided by the cosine function.

Q: Can you give an example of a trigonometric equation involving the secant function?

A: Yes, here is an example of a trigonometric equation involving the secant function:

\sec(\theta + 50^{\circ}) = 2

Q: How do you solve this equation?

A: To solve this equation, you need to use the reciprocal identity of the secant function, which is $\sec(\theta) = \frac{1}{\cos(\theta)}$ . You can then use the properties of the cosine function to solve the equation.

Q: What is the solution to the equation $\sec(\theta + 50^{\circ}) = 2$ ?

A: The solution to the equation $\sec(\theta + 50^{\circ}) = 2$ is $\theta = 10^{\circ}, 370^{\circ}$ .

Q: Can you explain the concept of the unit circle?

A: Yes, the unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a powerful tool for solving trigonometric equations.

Q: How do you use the unit circle to solve trigonometric equations?

A: To use the unit circle to solve trigonometric equations, you need to find the angle whose terminal side passes through the point on the unit circle that corresponds to the given trigonometric function.

Q: What is the relationship between the secant and cosine functions?

A: The secant function is the reciprocal of the cosine function. This means that the secant function is equal to 1 divided by the cosine function.

Q: Can you give an example of how to use the unit circle to solve a trigonometric equation?

A: Yes, here is an example of how to use the unit circle to solve a trigonometric equation:

Suppose we want to solve the equation $\sec(\theta) = 2$ . We can use the unit circle to find the angle whose terminal side passes through the point on the unit circle that corresponds to the secant function.

Q: How do you find the angle whose terminal side passes through the point on the unit circle that corresponds to the secant function?

A: To find the angle whose terminal side passes through the point on the unit circle that corresponds to the secant function, you need to use the properties of the unit circle and the definition of the secant function.

Q: What is the final solution to the equation $\sec(\theta + 50^{\circ}) = 2$ ?

A: The final solution to the equation $\sec(\theta + 50^{\circ}) = 2$ is $\theta = 10^{\circ}, 370^{\circ}$ .

Conclusion

In this article, we have solved the trigonometric equation $\sec(\theta + 50^{\circ}) = 2$ and found the solution set of $\theta$ where $90^{\circ} \leq \theta \leq 360^{\circ}$ . We have used the unit circle and the periodicity of the secant function to find the solutions. We have also answered some common questions about solving trigonometric equations involving the secant function.