From The Side View, A Gymnastics Mat Forms A Right Triangle With The Other Angles Measuring $60^{\circ}$ And $30^{\circ}$. The Gymnastics Mat Extends 5 Feet Across The Floor. How High Is The Mat Off The Ground?A.

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Introduction

In this article, we will explore a problem involving a right triangle formed by a gymnastics mat. The mat extends 5 feet across the floor and forms a right triangle with angles measuring 60° and 30°. Our goal is to find the height of the mat off the ground. We will use trigonometric ratios to solve this problem.

Understanding the Problem

A right triangle is a triangle with one angle measuring 90°. In this case, the right angle is formed by the mat and the floor. The other two angles are 60° and 30°. We are given the length of the side opposite the 30° angle, which is 5 feet. Our objective is to find the length of the side opposite the 90° angle, which is the height of the mat.

Using Trigonometric Ratios

To solve this problem, we will use trigonometric ratios. The sine, cosine, and tangent ratios are defined as follows:

  • Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse (the side opposite the 90° angle).
  • Cosine (cos): The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.

In this case, we will use the sine ratio to find the height of the mat. The sine ratio is defined as:

sin(θ) = opposite side / hypotenuse

where θ is the angle and the opposite side is the side opposite the angle.

Applying the Sine Ratio

We are given the length of the side opposite the 30° angle, which is 5 feet. We want to find the length of the side opposite the 90° angle, which is the height of the mat. Using the sine ratio, we can write:

sin(30°) = 5 / height

To find the height, we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by the height:

height * sin(30°) = 5

Now, we can divide both sides of the equation by sin(30°) to find the height:

height = 5 / sin(30°)

Evaluating the Sine of 30°

The sine of 30° is equal to 0.5. We can substitute this value into the equation:

height = 5 / 0.5

Simplifying the Equation

To simplify the equation, we can divide 5 by 0.5:

height = 10

Conclusion

In this article, we used trigonometric ratios to solve a problem involving a right triangle formed by a gymnastics mat. We found that the height of the mat off the ground is 10 feet.

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Architecture: Architects use trigonometry to design buildings and ensure that they are structurally sound.
  • Engineering: Engineers use trigonometry to design and build bridges, roads, and other infrastructure projects.
  • Physics: Physicists use trigonometry to describe the motion of objects and predict their trajectories.

Common Mistakes

When solving problems involving right triangles, it's essential to:

  • Use the correct trigonometric ratio: Make sure to use the correct ratio (sine, cosine, or tangent) for the given angle.
  • Evaluate the trigonometric function: Make sure to evaluate the trigonometric function (sine, cosine, or tangent) for the given angle.
  • Simplify the equation: Make sure to simplify the equation by performing the necessary operations (addition, subtraction, multiplication, or division).

Practice Problems

Here are some practice problems to help you reinforce your understanding of trigonometry:

  • Find the length of the side opposite the 60° angle in a right triangle with a hypotenuse of 10 feet.
  • Find the length of the side adjacent to the 30° angle in a right triangle with a hypotenuse of 15 feet.
  • Find the length of the side opposite the 90° angle in a right triangle with a side adjacent to the 30° angle of 8 feet.

Conclusion

Q: What is a right triangle?

A: A right triangle is a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse.

Q: What are the three main trigonometric ratios?

A: The three main trigonometric ratios are:

  • Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.

Q: How do I choose which trigonometric ratio to use?

A: To choose which trigonometric ratio to use, you need to identify the angle and the sides involved. If you know the length of the side opposite the angle and the length of the hypotenuse, use the sine ratio. If you know the length of the side adjacent to the angle and the length of the hypotenuse, use the cosine ratio. If you know the length of the side opposite the angle and the length of the side adjacent to the angle, use the tangent ratio.

Q: What is the sine of 30°?

A: The sine of 30° is equal to 0.5.

Q: What is the cosine of 60°?

A: The cosine of 60° is equal to 0.5.

Q: How do I evaluate the sine and cosine of angles?

A: To evaluate the sine and cosine of angles, you can use a calculator or a trigonometric table. Alternatively, you can use the unit circle to find the sine and cosine of angles.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle is used to find the sine and cosine of angles.

Q: How do I use the unit circle to find the sine and cosine of angles?

A: To use the unit circle to find the sine and cosine of angles, you need to locate the point on the unit circle corresponding to the angle. The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.

Q: What are some common mistakes to avoid when solving problems involving right triangles?

A: Some common mistakes to avoid when solving problems involving right triangles include:

  • Using the wrong trigonometric ratio: Make sure to use the correct ratio (sine, cosine, or tangent) for the given angle.
  • Evaluating the trigonometric function incorrectly: Make sure to evaluate the trigonometric function (sine, cosine, or tangent) for the given angle correctly.
  • Simplifying the equation incorrectly: Make sure to simplify the equation by performing the necessary operations (addition, subtraction, multiplication, or division).

Q: How can I practice solving problems involving right triangles?

A: You can practice solving problems involving right triangles by:

  • Solving practice problems: Try solving practice problems involving right triangles to reinforce your understanding of trigonometry.
  • Using online resources: Use online resources such as calculators and trigonometric tables to help you solve problems involving right triangles.
  • Working with a tutor or teacher: Work with a tutor or teacher to get help with solving problems involving right triangles.

Conclusion

In this article, we answered frequently asked questions about trigonometry and right triangles. We covered topics such as the definition of a right triangle, the three main trigonometric ratios, and how to choose which trigonometric ratio to use. We also discussed common mistakes to avoid when solving problems involving right triangles and provided tips for practicing solving problems involving right triangles.