Homework Answer Key Unit 8 Right Triangles And Trigonometry

Right triangles and trigonometry form the cornerstone of understanding geometric relationships and are critical in various fields, from engineering to navigation. A firm grasp of these concepts is essential for students progressing in mathematics. This homework answer key for Unit 8 is designed to provide comprehensive solutions, clarifying the principles of right triangles, trigonometric ratios, and their applications.

Understanding Right Triangles

A right triangle is a triangle that contains one angle of 90 degrees. The side opposite the right angle is known as the hypotenuse, the longest side of the triangle. The other two sides are called legs or cathetus. Understanding the relationships between these sides and the angles is fundamental to trigonometry.

Pythagorean Theorem

The Pythagorean Theorem is the backbone of right triangle calculations. It states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is expressed as:

a² + b² = c²

This theorem allows us to find the length of any side of a right triangle if the lengths of the other two sides are known.

Example 1: Find the length of the hypotenuse of a right triangle with legs of length 3 and 4.

Solution: Using the Pythagorean Theorem: a² + b² = c² 3² + 4² = c² 9 + 16 = c² 25 = c² c = √25 c = 5

Therefore, the length of the hypotenuse is 5.

Example 2: Find the length of the leg of a right triangle if the hypotenuse is 13 and the other leg is 5.

Solution: Using the Pythagorean Theorem: a² + b² = c² a² + 5² = 13² a² + 25 = 169 a² = 169 - 25 a² = 144 a = √144 a = 12

Therefore, the length of the leg is 12.

Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. This is used to verify whether a given triangle is a right triangle.

Example: Determine if a triangle with sides 7, 24, and 25 is a right triangle.

Solution: Check if 7² + 24² = 25² 49 + 576 = 625 625 = 625

Since the equation holds true, the triangle is a right triangle.

Trigonometric Ratios

Trigonometric ratios are functions that relate the angles of a right triangle to the ratios of its sides. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).

Sine (sin)

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

sin(θ) = Opposite / Hypotenuse

Cosine (cos)

The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

cos(θ) = Adjacent / Hypotenuse

Tangent (tan)

The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

tan(θ) = Opposite / Adjacent

Mnemonic: SOH CAH TOA

A common mnemonic to remember these ratios is SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Example Problems

Example 1: In a right triangle, the opposite side to angle θ is 6 and the hypotenuse is 10. Find sin(θ), cos(θ), and tan(θ). The adjacent side is 8 (using the Pythagorean Theorem).

Solution:

• sin(θ) = Opposite / Hypotenuse = 6 / 10 = 3 / 5
• cos(θ) = Adjacent / Hypotenuse = 8 / 10 = 4 / 5
• tan(θ) = Opposite / Adjacent = 6 / 8 = 3 / 4

Example 2: In a right triangle, angle α is 30 degrees and the hypotenuse is 12. Find the lengths of the opposite and adjacent sides.

Solution:

• sin(30°) = Opposite / 12. Since sin(30°) = 1/2, Opposite = 12 * (1/2) = 6
• cos(30°) = Adjacent / 12. Since cos(30°) = √3/2, Adjacent = 12 * (√3/2) = 6√3

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the measure of an angle when the ratio of the sides is known. The inverse functions are arcsine (sin⁻¹), arccosine (cos⁻¹), and arctangent (tan⁻¹).

Arcsine (sin⁻¹)

The arcsine of a value x is the angle whose sine is x.

θ = sin⁻¹(x)

Arccosine (cos⁻¹)

The arccosine of a value x is the angle whose cosine is x.

θ = cos⁻¹(x)

Arctangent (tan⁻¹)

The arctangent of a value x is the angle whose tangent is x.

θ = tan⁻¹(x)

Example Problems

Example 1: If sin(θ) = 0.5, find the measure of angle θ.

Solution: θ = sin⁻¹(0.5) = 30°

Example 2: If cos(θ) = √3/2, find the measure of angle θ.

Solution: θ = cos⁻¹(√3/2) = 30°

Example 3: If tan(θ) = 1, find the measure of angle θ.

Solution: θ = tan⁻¹(1) = 45°

Applications of Right Triangles and Trigonometry

Right triangles and trigonometry have numerous real-world applications, including:

Navigation

Trigonometry is used in navigation to determine distances and directions. By using angles of elevation and depression, navigators can calculate positions and chart courses.

Engineering

Engineers use trigonometric principles to design structures, calculate forces, and analyze stability. For example, they use trigonometry to determine the angles and lengths of support beams in bridges and buildings.

Surveying

Surveyors use trigonometry to measure land and create maps. They use angles and distances to calculate areas and elevations.

Physics

Trigonometry is used in physics to analyze vectors, motion, and forces. For example, it is used to calculate the components of a force acting at an angle.

Example Problems

Example 1: A ladder is leaning against a wall, forming a right triangle. The ladder is 10 feet long, and the base of the ladder is 6 feet from the wall. Find the angle of elevation of the ladder.

Solution: Let θ be the angle of elevation. cos(θ) = Adjacent / Hypotenuse = 6 / 10 = 0.6 θ = cos⁻¹(0.6) ≈ 53.13°

Therefore, the angle of elevation of the ladder is approximately 53.13 degrees.

Example 2: A building casts a shadow of 50 feet when the angle of elevation of the sun is 60 degrees. Find the height of the building.

Solution: Let h be the height of the building. tan(60°) = Opposite / Adjacent = h / 50 h = 50 * tan(60°) = 50 * √3 ≈ 86.6 feet

Therefore, the height of the building is approximately 86.6 feet.

Angle of Elevation and Depression

The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal line. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below the horizontal line.

Example Problems

Example 1: From the top of a cliff 100 feet high, the angle of depression to a boat is 25 degrees. Find the distance from the base of the cliff to the boat.

Solution: Let d be the distance from the base of the cliff to the boat. tan(25°) = Opposite / Adjacent = 100 / d d = 100 / tan(25°) ≈ 214.45 feet

Therefore, the distance from the base of the cliff to the boat is approximately 214.45 feet.

Example 2: An airplane is flying at an altitude of 3000 feet. The angle of elevation from the airport to the airplane is 15 degrees. Find the horizontal distance from the airport to the airplane.

Solution: Let d be the horizontal distance from the airport to the airplane. tan(15°) = Opposite / Adjacent = 3000 / d d = 3000 / tan(15°) ≈ 11196.15 feet

Therefore, the horizontal distance from the airport to the airplane is approximately 11196.15 feet.

Special Right Triangles

There are two special right triangles that are frequently encountered in trigonometry: the 45-45-90 triangle and the 30-60-90 triangle.

45-45-90 Triangle

In a 45-45-90 triangle, the two legs are congruent, and the angles are 45 degrees, 45 degrees, and 90 degrees. The ratio of the sides is 1:1:√2, where the legs are 1 and the hypotenuse is √2.

If a leg has length a, then the other leg also has length a, and the hypotenuse has length a√2.

30-60-90 Triangle

In a 30-60-90 triangle, the angles are 30 degrees, 60 degrees, and 90 degrees. The ratio of the sides is 1:√3:2, where the side opposite the 30-degree angle is 1, the side opposite the 60-degree angle is √3, and the hypotenuse is 2.

If the side opposite the 30-degree angle has length a, then the side opposite the 60-degree angle has length a√3, and the hypotenuse has length 2a.

Example Problems

Example 1: In a 45-45-90 triangle, the length of one leg is 5. Find the lengths of the other leg and the hypotenuse.

Solution: Since it's a 45-45-90 triangle, the other leg also has length 5. The hypotenuse has length 5√2.

Example 2: In a 30-60-90 triangle, the side opposite the 30-degree angle is 4. Find the lengths of the other two sides.

Solution: The side opposite the 60-degree angle has length 4√3, and the hypotenuse has length 2 * 4 = 8.

Solving Triangles

Solving a triangle means finding the measures of all its angles and the lengths of all its sides. To solve a right triangle, you typically need to know either:

• Two side lengths.
• One side length and one acute angle.

Example Problems

Example 1: Solve the right triangle where angle A = 30 degrees and side b (adjacent to angle A) = 10.

Solution:

1. Find angle B: Since it's a right triangle, angle C = 90 degrees. The sum of angles in a triangle is 180 degrees, so angle B = 180 - 90 - 30 = 60 degrees.
2. Find side a (opposite to angle A):
   • tan(A) = a/b
   • tan(30°) = a/10
   • a = 10 * tan(30°) = 10 * (1/√3) = 10√3 / 3 ≈ 5.77
3. Find side c (hypotenuse):
   • cos(A) = b/c
   • cos(30°) = 10/c
   • c = 10 / cos(30°) = 10 / (√3/2) = 20√3 / 3 ≈ 11.55

Therefore, angle B = 60 degrees, side a ≈ 5.77, and side c ≈ 11.55.

Example 2: Solve the right triangle where side a = 7 and side b = 9.

Solution:

1. Find side c (hypotenuse):
   • c² = a² + b²
   • c² = 7² + 9² = 49 + 81 = 130
   • c = √130 ≈ 11.40
2. Find angle A:
   • tan(A) = a/b = 7/9
   • A = tan⁻¹(7/9) ≈ 37.88 degrees
3. Find angle B:
   • B = 90 - A = 90 - 37.88 ≈ 52.12 degrees

Therefore, side c ≈ 11.40, angle A ≈ 37.88 degrees, and angle B ≈ 52.12 degrees.

Conclusion

Mastering right triangles and trigonometry is crucial for success in mathematics and various technical fields. This comprehensive guide, along with the provided examples, should equip you with the necessary tools to tackle homework problems and deepen your understanding of these fundamental concepts. Remember to practice regularly and apply these principles to real-world scenarios to reinforce your knowledge. Understanding these concepts will not only help you excel in your studies but also provide a solid foundation for future endeavors in science and engineering.