If One Third Of An Angle Is Equal To Its Supplement Then Find The Measure Of The Angle : A) 135° B)125° C) 110° D) 105°
Introduction
In trigonometry, angles are a fundamental concept that plays a crucial role in solving various mathematical problems. One of the essential concepts in trigonometry is the relationship between an angle and its supplement. In this article, we will explore the concept of an angle and its supplement, and then use this knowledge to solve a problem that involves finding the measure of an angle.
Understanding Angles and Their Supplements
An angle is a measure of the amount of rotation between two lines or planes. It is typically measured in degrees, with a full rotation being 360 degrees. The supplement of an angle is the angle that, when added to the original angle, equals 180 degrees. For example, if an angle is 60 degrees, its supplement is 120 degrees.
The Problem
The problem we will be solving is: "If one third of an angle is equal to its supplement, then find the measure of the angle." This problem involves using algebraic techniques to solve for the measure of the angle.
Step 1: Let's Define the Angle
Let's assume that the measure of the angle is x degrees. Since one third of the angle is equal to its supplement, we can write the equation:
(1/3)x = 180 - x
Step 2: Simplify the Equation
To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:
x = 540 - 3x
Step 3: Combine Like Terms
Next, we can combine like terms by adding 3x to both sides of the equation:
4x = 540
Step 4: Solve for x
Finally, we can solve for x by dividing both sides of the equation by 4:
x = 135
Conclusion
In this article, we used algebraic techniques to solve a problem that involved finding the measure of an angle. We defined the angle as x degrees, simplified the equation, combined like terms, and finally solved for x. The measure of the angle is 135 degrees.
Answer
The correct answer is:
- a) 135°
Why is this the correct answer?
This is the correct answer because when we solved the equation, we found that x = 135. This means that the measure of the angle is 135 degrees.
What is the relationship between the angle and its supplement?
The relationship between the angle and its supplement is that they add up to 180 degrees. In this problem, we were given that one third of the angle is equal to its supplement, which means that the angle and its supplement are related in a specific way.
What is the significance of this problem?
This problem is significant because it involves using algebraic techniques to solve for the measure of an angle. It also involves understanding the relationship between an angle and its supplement, which is an essential concept in trigonometry.
What are some real-world applications of this problem?
This problem has real-world applications in various fields, such as engineering, physics, and computer science. For example, in engineering, understanding the relationship between angles and their supplements is crucial for designing and building structures that are stable and secure.
What are some common mistakes to avoid when solving this problem?
Some common mistakes to avoid when solving this problem include:
- Not defining the angle as x degrees
- Not simplifying the equation
- Not combining like terms
- Not solving for x
What are some tips for solving this problem?
Some tips for solving this problem include:
- Define the angle as x degrees
- Simplify the equation
- Combine like terms
- Solve for x
What are some related problems that can be solved using similar techniques?
Some related problems that can be solved using similar techniques include:
- Finding the measure of an angle given its sine, cosine, or tangent
- Solving trigonometric equations involving multiple angles
- Finding the measure of an angle given its relationship to another angle
Conclusion
Q: What is the relationship between an angle and its supplement?
A: The relationship between an angle and its supplement is that they add up to 180 degrees. In other words, if an angle is x degrees, its supplement is 180 - x degrees.
Q: How do I define the angle in a trigonometric equation?
A: To define the angle in a trigonometric equation, you can let x be the measure of the angle. For example, if the equation involves the sine of the angle, you can write sin(x) = y, where y is the value of the sine function.
Q: What is the first step in solving a trigonometric equation?
A: The first step in solving a trigonometric equation is to simplify the equation by combining like terms and eliminating any fractions.
Q: How do I combine like terms in a trigonometric equation?
A: To combine like terms in a trigonometric equation, you can add or subtract the coefficients of the terms with the same variable. For example, if you have the equation 2sin(x) + 3sin(x) = 5, you can combine the like terms by adding the coefficients: 5sin(x) = 5.
Q: What is the next step in solving a trigonometric equation?
A: The next step in solving a trigonometric equation is to isolate the variable by dividing both sides of the equation by the coefficient of the variable. For example, if you have the equation 5sin(x) = 5, you can isolate the variable by dividing both sides by 5: sin(x) = 1.
Q: How do I solve for the variable in a trigonometric equation?
A: To solve for the variable in a trigonometric equation, you can use the inverse trigonometric function to find the value of the variable. For example, if you have the equation sin(x) = 1, you can use the inverse sine function to find the value of x: x = arcsin(1).
Q: What are some common mistakes to avoid when solving trigonometric equations?
A: Some common mistakes to avoid when solving trigonometric equations include:
- Not defining the angle as x degrees
- Not simplifying the equation
- Not combining like terms
- Not isolating the variable
- Not using the inverse trigonometric function to solve for the variable
Q: What are some tips for solving trigonometric equations?
A: Some tips for solving trigonometric equations include:
- Define the angle as x degrees
- Simplify the equation
- Combine like terms
- Isolate the variable
- Use the inverse trigonometric function to solve for the variable
Q: What are some related problems that can be solved using similar techniques?
A: Some related problems that can be solved using similar techniques include:
- Finding the measure of an angle given its sine, cosine, or tangent
- Solving trigonometric equations involving multiple angles
- Finding the measure of an angle given its relationship to another angle
Q: How do I apply trigonometric equations to real-world problems?
A: Trigonometric equations can be applied to real-world problems in various fields, such as engineering, physics, and computer science. For example, in engineering, trigonometric equations can be used to design and build structures that are stable and secure. In physics, trigonometric equations can be used to describe the motion of objects and predict their behavior.
Q: What are some common applications of trigonometric equations?
A: Some common applications of trigonometric equations include:
- Designing and building structures
- Describing the motion of objects
- Predicting the behavior of systems
- Solving problems in engineering, physics, and computer science
Q: How do I practice solving trigonometric equations?
A: To practice solving trigonometric equations, you can try solving problems on your own or using online resources, such as practice tests and worksheets. You can also work with a tutor or teacher to get help and feedback on your work.
Q: What are some resources for learning more about trigonometric equations?
A: Some resources for learning more about trigonometric equations include:
- Textbooks and online resources
- Practice tests and worksheets
- Tutors and teachers
- Online courses and tutorials
Conclusion
In conclusion, trigonometric equations are an essential tool for solving problems in various fields, such as engineering, physics, and computer science. By understanding the basics of trigonometric equations and practicing solving problems, you can develop the skills and knowledge you need to succeed in these fields.