Unit 7 Right Triangles And Trigonometry Homework 4 Answers Key

Unit 7 Right Triangles and Trigonometry Homework 4 Answers Key: A Comprehensive Guide

This comprehensive guide provides detailed explanations and solutions for common problems encountered in Unit 7, Homework 4, focusing on right triangles and trigonometry. We will cover fundamental concepts, essential formulas, and step-by-step solutions to various problem types. Understanding these concepts is crucial for mastering trigonometry and its applications in various fields.

Understanding Right Triangles

Before diving into the solutions, let's refresh our understanding of right triangles and the fundamental trigonometric ratios.

Right Triangle Definition: A right triangle is a triangle with one angle measuring 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called legs or cathetus.

Pythagorean Theorem: This theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically:

a² + b² = c²

where:

• a and b are the lengths of the legs
• c is the length of the hypotenuse

Trigonometric Ratios: These ratios relate the angles of a right triangle to the lengths of its sides. The three primary trigonometric ratios are:

• Sine (sin): sin(θ) = opposite / hypotenuse
• Cosine (cos): cos(θ) = adjacent / hypotenuse
• Tangent (tan): tan(θ) = opposite / adjacent

Where θ (theta) represents the angle.

Solving Right Triangle Problems: A Step-by-Step Approach

Let's tackle common problem types found in Unit 7, Homework 4, with detailed solutions. Remember to always label your triangle clearly, identifying the hypotenuse and the sides opposite and adjacent to the angle you are working with.

Problem Type 1: Finding Missing Sides Using Pythagorean Theorem

Problem Example: A right triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.

Solution:

1. Identify the knowns: a = 6 cm, b = 8 cm. We need to find c (the hypotenuse).
2. Apply the Pythagorean Theorem: 6² + 8² = c²
3. Solve for c: 36 + 64 = c² => 100 = c² => c = √100 = 10 cm

Therefore, the length of the hypotenuse is 10 cm.

Problem Type 2: Finding Missing Sides Using Trigonometric Ratios

Problem Example: In a right triangle, the hypotenuse is 10 cm and one angle is 30°. Find the length of the side opposite the 30° angle.

Solution:

1. Identify the knowns: Hypotenuse = 10 cm, θ = 30°. We need to find the opposite side.
2. Choose the appropriate trigonometric ratio: We know the hypotenuse and need the opposite side, so we use the sine function: sin(θ) = opposite / hypotenuse
3. Substitute the values: sin(30°) = opposite / 10 cm
4. Solve for the opposite side: opposite = 10 cm * sin(30°) = 10 cm * 0.5 = 5 cm

Therefore, the length of the side opposite the 30° angle is 5 cm.

Problem Type 3: Finding Missing Angles Using Trigonometric Ratios

Problem Example: In a right triangle, the legs have lengths 5 cm and 12 cm. Find the angle opposite the 12 cm leg.

Solution:

1. Identify the knowns: Opposite side = 12 cm, adjacent side = 5 cm. We need to find the angle (θ).
2. Choose the appropriate trigonometric ratio: We know the opposite and adjacent sides, so we use the tangent function: tan(θ) = opposite / adjacent
3. Substitute the values: tan(θ) = 12 cm / 5 cm = 2.4
4. Solve for θ: θ = arctan(2.4) (Use a calculator to find the inverse tangent)

Therefore, the angle opposite the 12 cm leg is approximately 67.4°

Problem Type 4: Solving Word Problems Involving Right Triangles

Problem Example: A ladder 15 meters long leans against a wall. The base of the ladder is 9 meters from the wall. Find the angle the ladder makes with the ground.

Solution:

1. Draw a diagram: Draw a right triangle with the ladder as the hypotenuse (15 m), the distance from the wall to the base of the ladder as one leg (9 m), and the height the ladder reaches on the wall as the other leg.
2. Identify the knowns: Hypotenuse = 15 m, adjacent side = 9 m. We need to find the angle (θ) between the ladder and the ground.
3. Choose the appropriate trigonometric ratio: We know the hypotenuse and the adjacent side, so we use the cosine function: cos(θ) = adjacent / hypotenuse
4. Substitute the values: cos(θ) = 9 m / 15 m = 0.6
5. Solve for θ: θ = arccos(0.6) (Use a calculator to find the inverse cosine)

Therefore, the angle the ladder makes with the ground is approximately 53.1°.

Advanced Concepts and Applications

Unit 7 might also introduce more advanced concepts, such as:

• Angles of Elevation and Depression: These are angles measured from a horizontal line of sight. Angles of elevation are measured upwards, while angles of depression are measured downwards. Many word problems will involve these concepts.
• Solving Oblique Triangles: While the focus of Unit 7 is on right triangles, you might encounter an introduction to solving triangles that are not right-angled. This usually involves the Law of Sines and the Law of Cosines.
• Applications of Trigonometry: Real-world applications of trigonometry are numerous, and homework problems might involve calculating heights of buildings, surveying land, or navigation problems.

Tips for Success in Trigonometry

• Master the Basics: Ensure a strong understanding of right triangles, Pythagorean theorem, and trigonometric ratios before tackling more complex problems.
• Practice Regularly: Consistent practice is key to mastering trigonometry. Work through numerous examples and practice problems.
• Use a Calculator: A scientific calculator is essential for solving trigonometric problems efficiently. Make sure you know how to use the sin, cos, tan, arcsin, arccos, and arctan functions.
• Draw Diagrams: Visualizing problems with a clear diagram can significantly simplify the problem-solving process. Label all sides and angles clearly.
• Check Your Answers: Always double-check your work and ensure your answers are reasonable within the context of the problem.

Conclusion

This comprehensive guide provides a solid foundation for understanding and solving problems related to right triangles and trigonometry, as typically encountered in Unit 7, Homework 4. By mastering the concepts discussed here and practicing regularly, you'll build a strong understanding of trigonometry and its wide-ranging applications. Remember, consistent effort and diligent practice are the keys to success in this essential area of mathematics. Don't hesitate to revisit these concepts and examples as needed to solidify your understanding. Good luck!