Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3v - 4w
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 34
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
A = 22°, b = 20 feet, c = 50 feet
Verified step by step guidance1
Identify the given elements of the triangle: angle \(A = 22^\circ\), side \(b = 20\) feet, and side \(c = 50\) feet. We need to find the area of the triangle.
Recall the formula for the area of a triangle when two sides and the included angle are known: \(\text{Area} = \frac{1}{2}bc \sin A\).
Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 20 \times 50 \times \sin 22^\circ\).
Calculate \(\sin 22^\circ\) using a calculator or trigonometric table to find the sine of the angle.
Multiply the values together to find the area, then round the result to the nearest whole number as requested.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Cosines
The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is useful for finding unknown sides or angles in any triangle, especially when two sides and the included angle are known. The formula is c² = a² + b² - 2ab cos(C).
Recommended video:
4:35
Intro to Law of Cosines
Area of a Triangle Using Two Sides and Included Angle
The area of a triangle can be calculated using two sides and the included angle with the formula: Area = 1/2 * b * c * sin(A). This method is efficient when two sides and the angle between them are given, avoiding the need to find the third side.
Recommended video:
4:02Calculating Area of SAS Triangles
Trigonometric Functions and Angle Measurement
Understanding sine and cosine functions and how to apply them to angles measured in degrees is essential. These functions relate angles to ratios of sides in right triangles and are fundamental in solving oblique triangles and calculating areas.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
In Exercises 35–36, the three given points are the vertices of a triangle. Solve each triangle, rounding lengths of sides to the nearest tenth and angle measures to the nearest degree.
A(0, 0), B(-3, 4), C(3, -1)
Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 1.4, b = 2.9, A = 142°
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Textbook Question
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = 3i - 2j, w = i - j
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Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||2u||
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Textbook Question
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 3j, w = -2i + 5j
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