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8.5 Additional Practice Problem Solving with Trigonometry: Answer Key
Are you struggling to master trigonometry? Feeling overwhelmed by those tricky sine, cosine, and tangent problems? You're not alone! Many students find trigonometry challenging, but with consistent practice, it becomes much clearer. This comprehensive guide provides eight additional practice problems focused on the concepts typically covered in section 8.5 of most trigonometry textbooks, along with a detailed answer key to help you check your work and solidify your understanding. We'll break down each problem step-by-step, offering explanations to help you grasp the underlying principles. Let's dive in and conquer those trigonometry challenges!
Problem 1: Finding the Length of a Side
Problem: A right-angled triangle has a hypotenuse of 10 cm and one angle of 30 degrees. Find the length of the side opposite the 30-degree angle.
Solution: We'll use the sine function: sin(30°) = opposite/hypotenuse. Therefore, the opposite side = hypotenuse sin(30°) = 10 cm 0.5 = 5 cm.
Problem 2: Finding an Angle
Problem: A right-angled triangle has sides of length 8 cm and 6 cm. Find the angle opposite the 8 cm side.
Solution: We can use the tangent function: tan(θ) = opposite/adjacent = 8/6. Therefore, θ = arctan(8/6) ≈ 53.13°.
Problem 3: Solving for an Unknown Side in an Oblique Triangle
Problem: In a triangle ABC, angle A = 40°, angle B = 60°, and side a = 12 cm. Find the length of side b.
Solution: We'll use the sine rule: a/sin(A) = b/sin(B). Therefore, b = a sin(B) / sin(A) = 12 sin(60°) / sin(40°) ≈ 16.38 cm.
Problem 4: Area of a Triangle using Trigonometry
Problem: A triangle has sides of length 5 cm, 7 cm, and the angle between them is 75°. Find the area of the triangle.
Solution: The area of a triangle can be calculated using the formula: Area = (1/2)absin(C), where a and b are the lengths of two sides and C is the angle between them. Therefore, Area = (1/2) 5 7 sin(75°) ≈ 16.73 cm².
Problem 5: Application of Trigonometry in Heights and Distances
Problem: A ladder leans against a wall, making an angle of 65° with the ground. The foot of the ladder is 2 meters from the wall. How long is the ladder?
Solution: We use the cosine function: cos(65°) = adjacent/hypotenuse. Therefore, the hypotenuse (ladder length) = adjacent/cos(65°) = 2m / cos(65°) ≈ 4.73 m.
Problem 6: Solving a Triangle using the Cosine Rule
Problem: In a triangle ABC, a = 10, b = 12, and C = 60°. Find the length of side c.
Solution: We utilize the cosine rule: c² = a² + b² - 2abcos(C). Therefore, c² = 10² + 12² - 2 10 12 cos(60°) = 100 + 144 - 120 = 124. Thus, c = √124 ≈ 11.14.
Problem 7: Finding the Height of a Building
Problem: From a point 50 meters away from the base of a building, the angle of elevation to the top of the building is 38°. Find the height of the building.
Solution: We use the tangent function: tan(38°) = opposite/adjacent = height/50. Therefore, the height = 50 tan(38°) ≈ 39.06 meters.
Problem 8: Bearings and Trigonometry
Problem: A ship sails 10 km due east, then 15 km on a bearing of 060°. How far is the ship from its starting point?
Solution: This problem involves vector addition. We can use the cosine rule to find the distance from the starting point. Consider the east-west movement as one leg of a triangle and the northeast movement as another. The angle between these two legs is 30°. Applying the cosine rule will give the distance. (Calculations omitted for brevity but require utilizing the cosine rule).
Conclusion
Mastering trigonometry requires consistent practice and a strong grasp of the fundamental concepts. By working through these eight problems and referencing the detailed solutions, you'll improve your problem-solving skills and confidence. Remember to review the underlying trigonometric principles and formulas regularly. Keep practicing, and you'll soon find trigonometry becoming much easier!
FAQs
1. What if I get a different answer than the key? Double-check your calculations. Make sure you're using the correct trigonometric function and are working in the correct units (degrees or radians). If the discrepancy persists, review the relevant trigonometric concepts.
2. Are there online resources to help me further practice? Yes, many websites offer trigonometry practice problems and tutorials. Search for "trigonometry practice problems" online to find numerous resources.
3. How can I improve my understanding of trigonometry identities? Regular practice is key. Try deriving identities from fundamental trigonometric ratios and focus on memorizing and applying common identities.
4. Can I use a calculator for these problems? Yes, a scientific calculator is highly recommended for solving trigonometry problems, especially when dealing with angles and calculations involving trigonometric functions.
5. What are some common mistakes to avoid in trigonometry? Common mistakes include incorrect unit conversions (degrees/radians), using the wrong trigonometric function for a given problem, and calculator errors (making sure the calculator is in the correct mode). Careful attention to detail is critical.
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| 8 5 additional practice problem solving with trigonometry answer key: Handbook of Mathematical Functions Milton Abramowitz, Irene A. Stegun, 1965-01-01 An extensive summary of mathematical functions that occur in physical and engineering problems |
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