Identify The Correct Trigonometry Formula To Use To Solve For { X $}$ Given The Following Triangle:- Side Adjacent To Angle { 55^\circ $}$ Is { 11 $}$- Opposite Side Is { X $}$- Angle Is { 55^\circ $}$
Introduction
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving a trigonometry problem involving a right-angled triangle. We will identify the correct trigonometry formula to use to solve for the unknown side, given the following triangle:
- Side adjacent to angle 55° is 11
- Opposite side is x
- Angle is 55°
Understanding the Triangle
Before we dive into the solution, let's understand the triangle. We have a right-angled triangle with an angle of 55°. The side adjacent to this angle is 11 units long, and the opposite side is x units long. We need to find the value of x.
Identifying the Correct Trigonometry Formula
To solve for x, we need to use the correct trigonometry formula. There are three basic trigonometry formulas that we can use to solve for the unknown side:
- Sine (sin) formula: sin(A) = opposite side / hypotenuse
- Cosine (cos) formula: cos(A) = adjacent side / hypotenuse
- Tangent (tan) formula: tan(A) = opposite side / adjacent side
In this case, we are given the adjacent side and the angle, so we can use the cosine formula to solve for x.
Applying the Cosine Formula
The cosine formula is cos(A) = adjacent side / hypotenuse. We can rearrange this formula to solve for the hypotenuse:
hypotenuse = adjacent side / cos(A)
We are given the adjacent side (11 units) and the angle (55°). We can plug these values into the formula:
hypotenuse = 11 / cos(55°)
To find the value of cos(55°), we can use a calculator or a trigonometry table. The value of cos(55°) is approximately 0.5736.
Solving for x
Now that we have the value of the hypotenuse, we can use the sine formula to solve for x:
sin(A) = opposite side / hypotenuse
We are given the angle (55°) and the hypotenuse (approximately 19.19 units). We can plug these values into the formula:
sin(55°) = x / 19.19
To find the value of sin(55°), we can use a calculator or a trigonometry table. The value of sin(55°) is approximately 0.8192.
Solving for x
Now that we have the value of sin(55°), we can solve for x:
x = sin(55°) x 19.19 x ≈ 0.8192 x 19.19 x ≈ 15.73
Conclusion
In this article, we identified the correct trigonometry formula to use to solve for the unknown side, given the following triangle:
- Side adjacent to angle 55° is 11
- Opposite side is x
- Angle is 55°
We used the cosine formula to find the value of the hypotenuse and then used the sine formula to solve for x. The value of x is approximately 15.73 units.
Common Trigonometry Formulas
Here are some common trigonometry formulas that you may find useful:
- Sine (sin) formula: sin(A) = opposite side / hypotenuse
- Cosine (cos) formula: cos(A) = adjacent side / hypotenuse
- Tangent (tan) formula: tan(A) = opposite side / adjacent side
- Pythagorean theorem: a² + b² = c² (where a and b are the legs of a right triangle, and c is the hypotenuse)
Tips and Tricks
Here are some tips and tricks that you may find useful when solving trigonometry problems:
- Use a calculator or a trigonometry table: These tools can help you find the values of trigonometric functions, such as sin, cos, and tan.
- Draw a diagram: Drawing a diagram of the triangle can help you visualize the problem and identify the correct trigonometry formula to use.
- Check your units: Make sure that your units are consistent throughout the problem.
- Use the Pythagorean theorem: The Pythagorean theorem can be used to find the length of the hypotenuse of a right triangle.
Real-World Applications
Trigonometry has numerous real-world applications, including:
- Navigation: Trigonometry is used in navigation to determine the position and direction of objects.
- Physics: Trigonometry is used in physics to describe the motion of objects and the forces that act upon them.
- Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
- Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.
Conclusion
In this article, we identified the correct trigonometry formula to use to solve for the unknown side, given the following triangle:
- Side adjacent to angle 55° is 11
- Opposite side is x
- Angle is 55°
Introduction
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some frequently asked questions about trigonometry.
Q: What is trigonometry?
A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation.
Q: What are the basic trigonometry formulas?
A: The basic trigonometry formulas are:
- Sine (sin) formula: sin(A) = opposite side / hypotenuse
- Cosine (cos) formula: cos(A) = adjacent side / hypotenuse
- Tangent (tan) formula: tan(A) = opposite side / adjacent side
Q: How do I use the sine formula to solve for the opposite side?
A: To use the sine formula to solve for the opposite side, you need to know the angle and the hypotenuse. You can rearrange the formula to solve for the opposite side:
opposite side = sin(A) x hypotenuse
Q: How do I use the cosine formula to solve for the adjacent side?
A: To use the cosine formula to solve for the adjacent side, you need to know the angle and the hypotenuse. You can rearrange the formula to solve for the adjacent side:
adjacent side = cos(A) x hypotenuse
Q: How do I use the tangent formula to solve for the opposite side?
A: To use the tangent formula to solve for the opposite side, you need to know the angle and the adjacent side. You can rearrange the formula to solve for the opposite side:
opposite side = tan(A) x adjacent side
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a formula that describes the relationship between the sides of a right triangle. It states that:
a² + b² = c²
where a and b are the legs of the triangle, and c is the hypotenuse.
Q: How do I use the Pythagorean theorem to solve for the hypotenuse?
A: To use the Pythagorean theorem to solve for the hypotenuse, you need to know the lengths of the two legs of the triangle. You can rearrange the formula to solve for the hypotenuse:
c = √(a² + b²)
Q: What are some common trigonometry problems?
A: Some common trigonometry problems include:
- Finding the length of the hypotenuse of a right triangle
- Finding the length of the opposite side of a right triangle
- Finding the length of the adjacent side of a right triangle
- Finding the angle of a right triangle
Q: How do I solve a trigonometry problem?
A: To solve a trigonometry problem, you need to:
- Read the problem carefully and identify the given information
- Choose the correct trigonometry formula to use
- Plug in the values into the formula
- Solve for the unknown side or angle
Q: What are some real-world applications of trigonometry?
A: Some real-world applications of trigonometry include:
- Navigation: Trigonometry is used in navigation to determine the position and direction of objects.
- Physics: Trigonometry is used in physics to describe the motion of objects and the forces that act upon them.
- Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
- Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.
Conclusion
In this article, we answered some frequently asked questions about trigonometry. We discussed the basic trigonometry formulas, how to use them to solve for the opposite side, adjacent side, and hypotenuse, and some common trigonometry problems. We also discussed some real-world applications of trigonometry.