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Chapter 11: Problem 84

Explain why no triangle \(A B C\) exists having \(A=103^{\circ}, a=12, b=13\)

Short Answer

Expert verified
This triangle cannot exist because \(\sin B\) exceeds 1, violating the range of the sine function.

Step by step solution

01

Establish the triangle's angle sum property

Start by recalling that the sum of angles in any triangle is always 180 degrees. Since angle \(A\) is 103 degrees, the sum of angles \(B\) and \(C\) must be 77 degrees.
02

Apply the Law of Sines

Using the Law of Sines, \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Plug in the values for \(a\), \(b\), and \(A\): \( \frac{12}{\sin 103^{\circ}} = \frac{13}{\sin B} \).
03

Solve the equation for \(\sin B\)

Calculate \(\sin 103^{\circ}\) and substitute it back to find \(\sin B\): \( \sin B = \frac{13 \times \sin 103^{\circ}}{12} \). Since \(103^{\circ}\) is an obtuse angle, \(\sin 103^{\circ}\) is positive and less than 1.
04

Check feasibility for \(\sin B\) within possible range

Calculate \(\sin B\): if \( \sin B > 1 \), then such a triangle cannot exist, due to the range limit of the sine function, which is [-1,1].
05

Conclude non-existence based on \(\sin B\)

Evaluating the value calculated in Step 3 reveals \(\sin B\) exceeds 1, indicating \(B\) is impossible to achieve, confirming such triangle \(ABC\) can't exist with given parameters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Law of Sines
The Law of Sines is a key concept when working with triangles, particularly in situations where some angles and sides are known, and others are not. It states that in any triangle, the ratio of the length of a side to the sine of its corresponding angle is constant across all three sides and angles of the triangle. Mathematically, this is expressed as:
  • \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
This formula is particularly useful when you know:
  • Two angles and one side (AAS or ASA) and need to find another side.
  • Two sides and a non-enclosed angle (SSA) and need to find an angle or the third side.
For example, given the problem's context:
  • We know angle \(A = 103^{\circ}\) and sides \(a = 12\), \(b = 13\).
  • Using the Law of Sines, \( \frac{12}{\sin 103^{\circ}} = \frac{13}{\sin B} \).
Solving this helps to understand if a triangle configuration is possible based on trigonometric limits. If the calculated sine of an angle is greater than 1, such a triangle isn't possible.
Triangle Inequality
One critical property of triangles is the Triangle Inequality Theorem. This theorem asserts that for any triangle, the sum of any two sides must be greater than the third side. The implications are:
  • For sides \(a\), \(b\), and \(c\) in a triangle, must always have: \( a + b > c \), \(a + c > b \), \(b + c > a \).
The Triangle Inequality is fundamental because it ensures that line segments can indeed form the sides of a triangle. For the triangle ABC specified, knowing this property can provide initial checks even before calculations:
  • With \(a = 12\) and \(b = 13\), you would anticipate another side \(c\) that adheres to these constraints to form a triangle.
If any inequalities are not satisfied, such a triangle can't exist.
Angle Sum Property
A fundamental principle of triangle geometry is the Angle Sum Property, which states that the sum of the internal angles of a triangle is always 180 degrees. This helps in finding out unknown angles when the other angles are known.
  • For example, in any triangle ABC, \(A + B + C = 180^{\circ}\).
In the original exercise, angle \(A\) is given as \(103^{\circ}\). Thus, \(B + C\) must sum up to \(77^{\circ}\) to satisfy the angle sum property:
  • \(B + C = 77^{\circ} \).
This property is pivotal when paired with other geometric or trigonometric rules to solve for unknowns, commonly while applying the Law of Sines or Cosines.

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Most popular questions from this chapter

The U.S. flag includes the colors red, white, and blue. Which color, red or white, is predominant? (Only \(18.73 \%\) of the total area is blue.) (Source: Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Banks, R... Princeton University Press.) (a) Let \(R\) denote the radius of the circumscribing circle of a five-pointed star appearing on the American flag. The star can be decomposed into 10 congruent triangles. In the figure below, \(r\) is the radius of the circumscribing circle of the pentagon in the interior of the star. Show that the area of the star is $$\begin{aligned}& \quad\quad\quad\quad\quad\quad A=\left[5 \frac{\sin A \sin B}{\sin (A+B)}\right] R^{2}\\\&\text { (Hint: }\left.\sin C=\sin \left[180^{\circ}-(A+B)\right]=\sin (A+B) .\right)\end{aligned}$$ (b) Angles \(A\) and \(B\) have values \(18^{\circ}\) and \(36^{\circ},\) respectively. Express the area of a star in terms of its radius \(R\) (c) To determine whether red or white is predominant, we consider a flag of width 10 inches, length 19 inches, length of each upper stripe 11.4 inches, and radius \(R\) of the circumscribing circle of each star 0.308 inch. The 13 stripes consist of six matching pairs of red and white stripes and one additional red, upper stripe. We must compare the area of a red, upper stripe with the total area of the 50 white stars. (i) Compute the area of the red, upper stripe. (ii) Compute the total area of the 50 white stars. (iii) Which color occupies the greatest area on the flag?

Solve each problem. Without actually performing the operations, state why the products $$\left[2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]\left[5\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$$ and $$\begin{array}{r}\left[2\left(\cos \left(-315^{\circ}\right)+i \sin \left(-315^{\circ}\right)\right)\right] \\ \left[5\left(\cos \left(-270^{\circ}\right)+i \sin \left(-270^{\circ}\right)\right)\right]\end{array}$$ are the same.

A projectile has been launched from the ground with an initial velocity of 88 feet per second. You are given parametric equations that model the path of the projectile. (a) Graph the parametric equations. (b) Approximate \(\theta\), the angle the projectile makes with the horizontal at launch, to the nearest tenth of a degree. (c) On the basis of your answer to part (b), write parametric equations for the projectile, using the cosine and sine functions. $$x=82.69265063 t, y=-16 t^{2}+30.09777261 t$$

Find a rectangular equation for each curve and graph the curve. $$x=2+\cos t, y=\sin t-1 ; 0 \leq t \leq 2 \pi$$

The graph of \(r=a \theta\) is an example of the spiral of Archimedes. With a calculator set to radian mode. use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=-1,0 \leq \theta \leq 12 \pi,[-40,40] \text { by }[-40,40]$$
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