39. A Cable Supports A Pole. It Is 15 M Long And Is Anchored 10 M Away. Find The Angle It Makes With The Ground.40. A Zipline Is 30 M Long And Has An Angle Of 25° With The Ground. Find The Height Of The Starting Point.

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Introduction

Right triangle trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of right triangles. It is a fundamental concept in mathematics and has numerous applications in various fields such as physics, engineering, and navigation. In this article, we will explore two problems related to right triangle trigonometry: finding the angle made by a cable with the ground and determining the height of the starting point of a zipline.

Problem 1: Finding the Angle Made by a Cable with the Ground

A cable supports a pole and is 15 m long. It is anchored 10 m away from the base of the pole. We need to find the angle it makes with the ground.

Step 1: Understand the Problem

We are given a right triangle with a hypotenuse (the cable) of length 15 m and one leg (the distance from the anchor to the base of the pole) of length 10 m. We need to find the angle between the cable and the ground.

Step 2: Draw a Diagram

Let's draw a diagram to visualize the problem.

  +---------------+
  |               |
  |  15m (cable)  |
  |               |
  +---------------+
  |               |
  |  10m (anchor) |
  |               |
  +---------------+
  |               |
  |  angle (θ)    |
  |               |
  +---------------+

Step 3: Use Trigonometric Ratios

We can use the sine, cosine, or tangent ratios to solve this problem. Let's use the sine ratio:

sin(θ) = opposite side (height of the pole) / hypotenuse (cable)

We don't know the height of the pole, but we can use the Pythagorean theorem to find it:

a^2 + b^2 = c^2

where a is the distance from the anchor to the base of the pole (10 m), b is the height of the pole (which we want to find), and c is the length of the cable (15 m).

Step 4: Solve for the Height of the Pole

Rearranging the Pythagorean theorem to solve for b, we get:

b = √(c^2 - a^2)

Plugging in the values, we get:

b = √(15^2 - 10^2) b = √(225 - 100) b = √125 b ≈ 11.18 m

Now that we have the height of the pole, we can use the sine ratio to find the angle:

sin(θ) = opposite side (height of the pole) / hypotenuse (cable) sin(θ) = 11.18 m / 15 m sin(θ) ≈ 0.746

Step 5: Find the Angle

Using a calculator or a trigonometric table, we can find the angle corresponding to the sine value:

θ ≈ arcsin(0.746) θ ≈ 48.59°

Therefore, the angle made by the cable with the ground is approximately 48.59°.

Problem 2: Finding the Height of the Starting Point of a Zipline

A zipline is 30 m long and has an angle of 25° with the ground. We need to find the height of the starting point.

Step 1: Understand the Problem

We are given a right triangle with a hypotenuse (the zipline) of length 30 m and an angle of 25° with the ground. We need to find the height of the starting point.

Step 2: Draw a Diagram

Let's draw a diagram to visualize the problem.

  +---------------+
  |               |
  |  30m (zipline) |
  |               |
  +---------------+
  |               |
  |  25° (angle)  |
  |               |
  +---------------+
  |               |
  |  height (h)   |
  |               |
  +---------------+

Step 3: Use Trigonometric Ratios

We can use the sine, cosine, or tangent ratios to solve this problem. Let's use the sine ratio:

sin(θ) = opposite side (height of the starting point) / hypotenuse (zipline)

We know the angle (25°) and the length of the zipline (30 m), so we can plug in the values:

sin(25°) = opposite side (height of the starting point) / 30 m

Step 4: Solve for the Height of the Starting Point

Using a calculator or a trigonometric table, we can find the sine value:

sin(25°) ≈ 0.4226

Now we can solve for the height of the starting point:

height (h) = sin(25°) × 30 m h ≈ 0.4226 × 30 m h ≈ 12.68 m

Therefore, the height of the starting point of the zipline is approximately 12.68 m.

Conclusion

Frequently Asked Questions

Q: What is right triangle trigonometry?

A: Right triangle trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of right triangles. It is a fundamental concept in mathematics and has numerous applications in various fields such as physics, engineering, and navigation.

Q: What are the three main trigonometric ratios?

A: The three main trigonometric ratios are:

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

Q: How do I use trigonometry to solve problems?

A: To use trigonometry to solve problems, you need to:

  1. Draw a diagram: Draw a diagram of the problem to visualize the situation.
  2. Identify the trigonometric ratio: Identify the trigonometric ratio that is relevant to the problem.
  3. Plug in the values: Plug in the values given in the problem into the trigonometric ratio.
  4. Solve for the unknown: Solve for the unknown value using the trigonometric ratio.

Q: What are some common applications of trigonometry?

A: Some common applications of trigonometry include:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions.
  • Physics: Trigonometry is used in physics to calculate the motion of objects.
  • Engineering: Trigonometry is used in engineering to design and build structures.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Q: How do I use a calculator to solve trigonometry problems?

A: To use a calculator to solve trigonometry problems, you need to:

  1. Enter the values: Enter the values given in the problem into the calculator.
  2. Choose the trigonometric function: Choose the trigonometric function that is relevant to the problem.
  3. Press the calculate button: Press the calculate button to get the answer.

Q: What are some common mistakes to avoid when using trigonometry?

A: Some common mistakes to avoid when using trigonometry include:

  • Not drawing a diagram: Not drawing a diagram can make it difficult to visualize the problem.
  • Not identifying the trigonometric ratio: Not identifying the trigonometric ratio can lead to incorrect solutions.
  • Not plugging in the values correctly: Not plugging in the values correctly can lead to incorrect solutions.
  • Not solving for the unknown correctly: Not solving for the unknown correctly can lead to incorrect solutions.

Q: How can I practice trigonometry?

A: You can practice trigonometry by:

  • Solving problems: Solving problems is a great way to practice trigonometry.
  • Using online resources: Using online resources such as Khan Academy and Mathway can help you practice trigonometry.
  • Working with a tutor: Working with a tutor can help you practice trigonometry and get feedback on your work.

Conclusion

In this article, we have answered some frequently asked questions about right triangle trigonometry. We have covered topics such as the three main trigonometric ratios, how to use trigonometry to solve problems, and common applications of trigonometry. We have also provided tips on how to practice trigonometry and avoid common mistakes. We hope this article has been helpful in your understanding of right triangle trigonometry.