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Chapter 6: Problem 64

What does it mean to solve an oblique triangle?

Short Answer

Expert verified
Solving an oblique triangle involves finding all sides and angles using the law of sines and/or the law of cosines. The former is used when you know two angles and a side or two sides and an angle opposite one of them, and the latter when you know the lengths of two sides and the size of their included angle, or the lengths of all the sides and want to calculate an angle.

Step by step solution

01

Understand the structure of an oblique triangle

An oblique triangle does not contain a right angle (90 degrees). It can either be an acute triangle where all angles are more than 0 but less than 90 degrees, or it is an obtuse triangle where one angle is greater than 90 degrees but less than 180 degrees.
02

The Laws to solve an oblique triangle

There are two rules you use to solve an oblique triangle. The law of sines allows you to find an unknown side or angle in a triangle if you know two angles and one side, or if you know two sides and an angle opposite one of them. The law of cosines allows you to calculate the length of an unknown side of a triangle if you know the lengths of the other two sides and the size of the angle between them, or if you know the lengths of all three sides and want to find an angle.
03

Applying the Laws

Both laws use the ratios created by the sides and angles in a triangle. In the law of sines the ratio between the length of a side and the sine of its opposite angle is the same for all three sides (a/sin(A) = b/sin(B) = c/sin(C)). In the law of cosines, the square of one side equals the sum of the squares of the other two sides minus twice their product times the cosine of the included angle (c^2 = a^2+b^2-2ab*cos(C)).

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$(1+i)(2+2 i)$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=3 \mathbf{i}, \quad \mathbf{w}=-4 \mathbf{i}$$

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}-10 \mathbf{j}$$

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=3 \cos \frac{5 \pi}{3} \mathbf{i}+3 \sin \frac{5 \pi}{3} \mathbf{j}, \quad \mathbf{w}=2 \cos \pi \mathbf{i}+2 \sin \pi \mathbf{j}$$

Graph the spiral \(r=\frac{1}{\theta} .\) Use a [-1.6,1.6,1] by [-1,1,1] viewing rectangle. Let \(\theta \min =0\) and \(\theta \max =2 \pi,\) then \(\theta \min =0\) and \(\theta \max =4 \pi,\) and finally \(\theta \min =0\) and \(\theta \max =8 \pi\)
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