Textbook Question
Use the figure to find each vector: - u. Use vector notation as in Example 4.
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 33
Solve each triangle. See Examples 2 and 3.
A = 112.8°, b = 6.28 m, c = 12.2 m
Verified step by step guidance1
Identify the given elements of the triangle: angle \(A = 112.8^\circ\), side \(b = 6.28\) m, and side \(c = 12.2\) m. We need to find the remaining angles \(B\) and \(C\), and side \(a\).
Use the Law of Cosines to find side \(a\). The Law of Cosines formula is:
\[a^2 = b^2 + c^2 - 2bc \cos A\]
Substitute the known values of \(b\), \(c\), and \(A\) into this formula.
Calculate \(a\) by taking the square root of the result from the Law of Cosines step:
\[a = \sqrt{b^2 + c^2 - 2bc \cos A}\]
Use the Law of Sines to find one of the unknown angles, for example angle \(B\). The Law of Sines states:
\[\frac{\sin A}{a} = \frac{\sin B}{b}\]
Rearrange to solve for \(\sin B\):
\[\sin B = b \times \frac{\sin A}{a}\]
Calculate angle \(B\) by taking the inverse sine (arcsin) of \(\sin B\). Then find angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\):
\[C = 180^\circ - A - B\]
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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 especially useful for solving triangles when two sides and the included angle are known. The formula is c² = a² + b² - 2ab cos(C), allowing calculation of unknown sides or angles.
Recommended video:
4:35
Intro to Law of Cosines
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. Knowing one angle and two sides allows you to find the remaining angles by subtracting the known angles from 180°, which is essential for fully solving the triangle.
Recommended video:
4:47Sum and Difference of Tangent
Law of Sines
The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant in a triangle: a/sin(A) = b/sin(B) = c/sin(C). This law helps find unknown angles or sides when given an angle-side pair and another side or angle.
Recommended video:
4:27Intro to Law of Sines
Related Practice
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