Textbook Question
A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane.
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 55
Find the area of each triangle ABC.
A = 56.80°, b = 32.67 in., c = 52.89 in.
Verified step by step guidance1
Identify the given elements: angle \(A = 56.80^\circ\), side \(b = 32.67\) inches, and side \(c = 52.89\) inches. 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: \(\text{Area} = \frac{1}{2}bc \sin A\).
Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 32.67 \times 52.89 \times \sin(56.80^\circ)\).
Calculate \(\sin(56.80^\circ)\) using a calculator or trigonometric table, ensuring your calculator is in degree mode.
Multiply the values together to find the area: first multiply the two sides, then multiply by the sine of the angle, and finally multiply by \(\frac{1}{2}\).
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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), which helps in determining missing elements in oblique triangles.
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Intro to Law of Cosines
Area of a Triangle Using Two Sides and Included Angle
The area of a triangle can be calculated using the formula (1/2)bc sin(A), where b and c are two sides and A is the included angle between them. This method is especially useful when the height is unknown but two sides and the included angle are given, allowing direct computation of the area.
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 extend to oblique triangles, enabling calculation of unknown sides or areas using given angles and side lengths.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Find the dot product for each pair of vectors.
4i, 5i - 9j
Textbook Question
Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈2, 1〉, 〈-3, 1〉
Textbook Question
A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57° 40′. A wind is blowing from the south at 27.1 mph. Find the bearing she should fly, and find the plane's ground speed.
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Textbook Question
Find the area of each triangle ABC.
A = 59.80°, b = 15.00 cm, C = 53.10°
Textbook Question
Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈4, 0〉, 〈2, 2〉