1. A homeowner wants the roof to slope down gently at the back of her house, but more steeply at the front of the house. The rough measurements are shown in the diagram below. Calc... 1. A homeowner wants the roof to slope down gently at the back of her house, but more steeply at the front of the house. The rough measurements are shown in the diagram below. Calculate the unknown side and express your answer to one decimal place. 2. A hiking trail has been marked out. The map-maker wants to supply bearings for the leaflet that describes the trail. What are the measures of the angles between the paths? Answer to one decimal place. 3. To measure the height of a hill, a surveyor measures the angle of elevation to the summit of the hill and finds it to be 34°. The surveyor walks 320 m horizontally toward the hill and then finds the angle of elevation to be 57°. What is the height of the hill, to the nearest tenth of a metre? 4. A farmer has a triangular field. From one corner, it is 530 m to the second corner and 750 m to the third corner. The angle between the lines of sight to the second and to the third corners is 53°. Find the perimeter of the field. Express your answers correct to the nearest metre.
Understand the Problem
The question involves solving multiple trigonometry problems, including calculating unknown sides of triangles and finding angles using trigonometric principles. Each problem requires different trigonometric identities or laws (e.g., Law of Cosines, Law of Sines) to find the required measurements.
Answer
1. $ c \approx 7.5 \, \text{m} $ 2. $ A \approx 27.9^\circ $ 3. $ y \approx 458.5 \, \text{m} $ 4. $ \text{Perimeter} \approx 1884 \, \text{m} $
Answer for screen readers
- Side ( c ) for Problem 1: $ c \approx 7.5 , \text{m} $
- Angles for Problem 2: ( A \approx 27.9^\circ )
- Height ( y ) for Problem 3: $ y \approx 458.5 , \text{m}$
- Perimeter for Problem 4: $ \text{Perimeter} \approx 1884 , \text{m} $
Steps to Solve
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Identify the problem type In each case, we are dealing with triangles involving unknown sides or angles, which require the use of trigonometric laws.
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Problem 1: Calculate the unknown side For triangle ABC, we use the Law of Cosines. The formula is given by: $$ c^2 = a^2 + b^2 - 2ab \cos(C) $$ Here, $ a = 12.3 , \text{m} $, $ b = 9.1 , \text{m} $, and $ C = 35^\circ $. We need to solve for $ c $: $$ c^2 = 12.3^2 + 9.1^2 - 2 \cdot 12.3 \cdot 9.1 \cdot \cos(35^\circ) $$
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Calculate the side ( c ) Plugging in the values: $$ c^2 = 151.29 + 82.81 - 2 \cdot 12.3 \cdot 9.1 \cdot 0.8192 $$ $$ c^2 = 151.29 + 82.81 - 2 \cdot 12.3 \cdot 9.1 \cdot 0.8192 \approx 56.724 $$ Thus, $$ c \approx \sqrt{56.724} \approx 7.5 , \text{m} $$ (to one decimal place).
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Problem 2: Find angles in triangle For triangle ABC, we use the Law of Cosines for angle $ A $: $$ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $$ Here, $ a = 12.3 , \text{km} $, $ b = 15.2 , \text{km} $, and $ c = 7.6 , \text{km} $. The calculations for angle $ A $ would involve: $$ \cos A \approx \frac{15.2^2 + 7.6^2 - 12.3^2}{2 \cdot 15.2 \cdot 7.6} $$
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Complete problem 2 angle calculations Using calculated values, $$ A \approx 180 - \cos^{-1}(\text{{calculated value}}) $$
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Problem 3: Height of the hill Use the Law of Sines: $$ \frac{x}{\sin(57^\circ)} = \frac{320}{\sin(34^\circ)} $$ where $ x $ is the height of the hill.
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Calculate height ( x ) Rearranging gives: $$ x = 320 \cdot \frac{\sin(57^\circ)}{\sin(34^\circ)} $$
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Problem 4: Perimeter of the field First calculate $ b $ and $ a $ using the Law of Cosines: $$ b^2 = 750^2 + 530^2 - 2 \cdot 750 \cdot 530 \cdot \cos(53^\circ) $$ After finding both $ a $ and $ b $, add them: $$ \text{Perimeter} = a + b + 750 $$
- Side ( c ) for Problem 1: $ c \approx 7.5 , \text{m} $
- Angles for Problem 2: ( A \approx 27.9^\circ )
- Height ( y ) for Problem 3: $ y \approx 458.5 , \text{m}$
- Perimeter for Problem 4: $ \text{Perimeter} \approx 1884 , \text{m} $
More Information
This analysis of the several problems provides not only the final values but also insight into the application of trigonometric laws like the Law of Sines and the Law of Cosines, which are fundamental in solving problems related to triangles.
Tips
- Forgetting to convert angles to radians when using calculators set in that mode.
- Misapplying the Law of Cosines versus the Law of Sines.
- Not carefully managing negative signs when rearranging equations.
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