If An Angle Is $15^{\circ}$ More Than Its Complement, What Is The Angle?A. $37.5^{\circ}$ B. $45^{\circ}$ C. $52.5^{\circ}$ D. $65^{\circ}$

Introduction

In this article, we will delve into the world of trigonometry and explore a problem that involves finding an angle based on its relationship with its complement. The problem states that an angle is $15^{\circ}$ more than its complement, and we need to determine the value of this angle. We will break down the solution into manageable steps, using mathematical concepts and formulas to arrive at the correct answer.

Understanding Complementary Angles

Before we dive into the problem, let's take a moment to understand what complementary angles are. Complementary angles are two angles whose sum is $90^{\circ}$. In other words, if we have two angles, $A$ and $B$, and they are complementary, then $A + B = 90^{\circ}$.

The Problem

Now that we have a basic understanding of complementary angles, let's move on to the problem at hand. We are given that an angle, which we will call $x$, is $15^{\circ}$ more than its complement. Mathematically, this can be represented as:

$$x = 90^{\circ} - x + 15^{\circ}$$

Simplifying the Equation

To solve for $x$, we need to simplify the equation. Let's start by combining like terms:

$$x = 105^{\circ} - x$$

Next, we can add $x$ to both sides of the equation to get:

$$2x = 105^{\circ}$$

Solving for x

Now that we have the equation $2x = 105^{\circ}$, we can solve for $x$ by dividing both sides of the equation by $2$:

$$x = \frac{105^{\circ}}{2}$$

Calculating the Value of x

To find the value of $x$, we can perform the division:

$$x = 52.5^{\circ}$$

Conclusion

In this article, we have solved a problem that involves finding an angle based on its relationship with its complement. We started by understanding the concept of complementary angles and then moved on to the problem at hand. By simplifying the equation and solving for $x$, we arrived at the correct answer, which is $52.5^{\circ}$.

Answer

The correct answer is:

• C. $52.5^{\circ}$

Additional Tips and Tricks

• When solving problems involving complementary angles, make sure to remember that the sum of the two angles is always $90^{\circ}$.
• When simplifying equations, look for opportunities to combine like terms.
• When solving for a variable, make sure to isolate the variable on one side of the equation.

Frequently Asked Questions

• Q: What is the definition of complementary angles? A: Complementary angles are two angles whose sum is $90^{\circ}$.
• Q: How do I simplify an equation? A: To simplify an equation, look for opportunities to combine like terms.
• Q: How do I solve for a variable? A: To solve for a variable, make sure to isolate the variable on one side of the equation.

Related Topics

• Trigonometry
• Complementary angles
• Solving equations

References

• [1] "Trigonometry" by Michael Corral
• [2] "Algebra and Trigonometry" by James Stewart

About the Author

Introduction

In our previous article, we explored the concept of complementary angles and solved a problem involving an angle that is $15^{\circ}$ more than its complement. In this article, we will answer some of the most frequently asked questions related to complementary angles and trigonometry.

Q&A

Q: What is the definition of complementary angles?

A: Complementary angles are two angles whose sum is $90^{\circ}$. In other words, if we have two angles, $A$ and $B$, and they are complementary, then $A + B = 90^{\circ}$.

Q: How do I find the complement of an angle?

A: To find the complement of an angle, subtract the angle from $90^{\circ}$. For example, if we have an angle of $30^{\circ}$, its complement is $90^{\circ} - 30^{\circ} = 60^{\circ}$.

Q: What is the relationship between complementary angles and trigonometry?

A: Complementary angles are an important concept in trigonometry, as they are used to solve problems involving right triangles. In a right triangle, the two acute angles are complementary, and their sum is always $90^{\circ}$.

Q: How do I solve problems involving complementary angles?

A: To solve problems involving complementary angles, follow these steps:

1. Identify the complementary angles.
2. Set up an equation using the fact that the sum of the two angles is $90^{\circ}$.
3. Solve the equation for the unknown angle.

Q: What is the difference between complementary angles and supplementary angles?

A: Complementary angles are two angles whose sum is $90^{\circ}$, while supplementary angles are two angles whose sum is $180^{\circ}$. For example, if we have two angles of $60^{\circ}$ and $30^{\circ}$, they are complementary, while if we have two angles of $120^{\circ}$ and $60^{\circ}$, they are supplementary.

Q: How do I use trigonometric functions to solve problems involving complementary angles?

A: To use trigonometric functions to solve problems involving complementary angles, follow these steps:

1. Identify the trigonometric function that is relevant to the problem.
2. Set up an equation using the trigonometric function and the fact that the two angles are complementary.
3. Solve the equation for the unknown angle.

Q: What are some common mistakes to avoid when working with complementary angles?

A: Some common mistakes to avoid when working with complementary angles include:

• Failing to identify the complementary angles.
• Setting up the wrong equation.
• Not solving for the unknown angle.

Conclusion

In this article, we have answered some of the most frequently asked questions related to complementary angles and trigonometry. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of these important concepts.

Additional Resources

• [1] "Trigonometry" by Michael Corral
• [2] "Algebra and Trigonometry" by James Stewart
• [3] "Mathematics for Dummies" by Mary Jane Sterling

About the Author

The author of this article is a mathematics enthusiast with a passion for solving problems and sharing knowledge with others.

Related Topics

• Trigonometry
• Complementary angles
• Supplementary angles
• Trigonometric functions

References

• [1] "Trigonometry" by Michael Corral
• [2] "Algebra and Trigonometry" by James Stewart
• [3] "Mathematics for Dummies" by Mary Jane Sterling