Textbook Question
A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?
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Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 51
Find the area of each triangle ABC.
A = 42.5°, b = 13.6 m, c = 10.1 m
Verified step by step guidance1
Identify the given elements of the triangle: angle \(A = 42.5^\circ\), side \(b = 13.6\) m, and side \(c = 10.1\) m. We need to find the area of triangle \(ABC\).
Recall the formula for the area of a triangle when two sides and the included angle are known: \(\text{Area} = \frac{1}{2} \times b \times c \times \sin(A)\).
Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 13.6 \times 10.1 \times \sin(42.5^\circ)\).
Calculate \(\sin(42.5^\circ)\) using a calculator or trigonometric table to find the sine of the given angle.
Multiply the values together: half of the product of sides \(b\) and \(c\), then multiply by \(\sin(42.5^\circ)\) to find the area of the triangle.
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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 the sides of a triangle to the cosine of one of its angles. It is useful for finding an unknown side or angle when two sides and the included angle are known. The formula is c² = a² + b² - 2ab cos(C), helping to determine missing elements in non-right triangles.
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 especially useful when the height is unknown but two sides and the angle between them are given, allowing direct computation of the area.
Recommended video:
4:02Calculating Area of SAS Triangles
Sine Function in Trigonometry
The sine function relates an angle of a triangle to the ratio of the length of the opposite side over the hypotenuse in a right triangle. In non-right triangles, sine is used in formulas like the area formula and the Law of Sines, making it essential for solving problems involving angles and side lengths.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Find the area of each triangle ABC.
B = 124.5°, a = 30.4 cm, c = 28.4 cm
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Textbook Question
Solve each problem. See Examples 5 and 6.
Bearing and Ground Speed of a Plane An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from a direction of 114.0°. Find the resulting bearing and ground speed of the plane.
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Textbook Question
Find the force required to keep a 75-lb sled from sliding down an incline that makes an angle of 27° with the horizontal. (Assume there is no friction.)
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Textbook Question
One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.
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Textbook Question
Solve each problem. See Examples 5 and 6.
Distance and Direction of a Motorboat A motorboat sets out in the direction N 80° 00′ E. The speed of the boat in still water is 20.0 mph. If the current is flowing directly south, and the actual direction of the motorboat is due east, find the speed of the current and the actual speed of the motorboat.
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