Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i
1
views
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 4, Problem 30
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 = 95, c = 125, A = 49°
Verified step by step guidance1
Identify the given elements: side \(a = 95\), side \(c = 125\), and angle \(A = 49^\circ\). Note that angle \(A\) is opposite side \(a\).
Use the Law of Sines to find angle \(C\) or to check the number of possible triangles. The Law of Sines states: \(\frac{a}{\sin A} = \frac{c}{\sin C}\).
Rearrange the Law of Sines to solve for \(\sin C\): \(\sin C = \frac{c \cdot \sin A}{a}\).
Calculate \(\sin C\) using the given values (do not find the final value yet). Then analyze the value of \(\sin C\) to determine the number of possible triangles: if \(\sin C > 1\), no triangle; if \(\sin C = 1\), one right triangle; if \(0 < \sin C < 1\), two possible triangles (since \(\sin\) is positive in two quadrants).
If one or two triangles exist, find angle \(C\) by taking \(\arcsin(\sin C)\) for the first solution, and for the second solution use \(180^\circ - C\). Then find angle \(B\) using the triangle angle sum \(B = 180^\circ - A - C\), and finally find side \(b\) using the Law of Sines: \(\frac{b}{\sin B} = \frac{a}{\sin A}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Sines
The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is essential for solving triangles when given two sides and a non-included angle (SSA), as it helps find unknown angles or sides.
Recommended video:
4:27
Intro to Law of Sines
Ambiguous Case of SSA Triangles
The SSA configuration can produce zero, one, or two possible triangles, known as the ambiguous case. This depends on the given side lengths and angle, particularly the height relative to the known side. Understanding this helps determine how many triangles satisfy the given measurements.
Recommended video:
9:50Solving SSA Triangles ("Ambiguous" Case)
Triangle Solution and Rounding
After determining the number of triangles, solving involves calculating missing sides and angles using trigonometric laws. Final answers should be rounded appropriately—angles to the nearest degree and sides to the nearest tenth—to provide clear, practical results consistent with the problem's instructions.
Recommended video:
5:3530-60-90 Triangles
Related Practice
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
-4w
1
views
Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4j
Textbook Question
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
||-2v||
Textbook Question
In Exercises 31–32, find the unit vector that has the same direction as the vector v.
v = -i + 2j
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3w + 2v
1
views