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Chapter 8: Problem 6

Solve the triangle, if possible. (triangle can't copy) $$C=22.28^{\circ}, a=25.4 \mathrm{cm}, b=73.8 \mathrm{cm}$$

Short Answer

Expert verified
Triangle is not possible because \( \sin B > 1 \).

Step by step solution

01

Identify Given Information

Given: Angle C = 22.28掳, Side a = 25.4 cm, Side b = 73.8 cm. We need to find the other angles A and B, and side c.
02

Use the Law of Sines

The Law of Sines states: \ \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] \ Using the given values: \ \[ \frac{25.4}{\sin A} = \frac{73.8}{\sin B} = \frac{c}{\sin 22.28^\circ} \]
03

Find Angle A

Rearrange the equation to solve for \(\sin A\): \ \[ \sin A = \sin 22.28^\circ \cdot \frac{25.4}{c} \] \ To find A, we must solve for c first.
04

Solve for Side c

Since \( c = \frac{b \cdot \sin 22.28^\circ}{\sin B} \), we first need to find another angle. Use \( \sin B = \frac{b \cdot \sin A}{a} \), and solve for \( B \).
05

Determine Angle B

Use the identity: \ \[ \sin B = \frac{73.8 \cdot \sin 22.28^\circ}{25.4} \] \ Calculate the value of \( \sin B = \frac{73.8 \cdot 0.3782}{25.4} \approx 1.099 \). This value is greater than 1, thus it鈥檚 not possible for a valid triangle.

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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 fundamental principle in trigonometry that relates the sides and angles of a triangle. It's particularly useful for solving triangles when given either two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).

The formula for the Law of Sines is: \ \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

This equation tells us that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides and angles.

To use the Law of Sines:
  • Identify and label the known angles and sides.
  • Set up the ratios based on the known values.
  • Solve for the unknown angles and sides using algebraic manipulation and trigonometric functions.
Its versatility makes it an essential tool for trigonometry problems involving non-right triangles.
Triangle Properties
Understanding triangle properties is crucial in trigonometry. Here are some key properties:

  • **Triangles are 2D shapes with three sides and three angles.**
  • **The sum of the interior angles of any triangle is always 180 degrees.**
  • **The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.**

There are different types of triangles based on sides and angles:
  • **Scalene Triangle:** All sides and angles are different.
  • **Isosceles Triangle:** Two sides are equal, and the angles opposite those sides are equal.
  • **Equilateral Triangle:** All three sides and angles are equal.

    • Also, the angles can classify triangles as:
    • **Acute Triangle:** All angles are less than 90 degrees.
    • **Right Triangle:** One angle is exactly 90 degrees.
    • **Obtuse Triangle:** One angle is greater than 90 degrees.
    These properties are beneficial when solving any triangle problem, whether you're using the Law of Sines, Law of Cosines, or other trigonometric methods.
Angle Calculation
Calculating angles in a triangle involves using various trigonometric principles. Here are the steps for calculating angles:

  • **Identify the given information:** List all known sides and angles.
  • **Use relevant trigonometric formulas:** Based on known values, choose the appropriate formula.
  • **Apply the Law of Sines or Law of Cosines:** These laws relate the angles to the sides of the triangle.

    • \ \( \text{For example, if you need to find angle A when two sides a and b are known: } \ \sin A = \sin C \cdot \frac{a}{c} \).

  • **Check the sum of angles:** Ensure the sum is 180 degrees to verify the calculations.

In the given problem, we found:

\ \( \text{Angle B:} \ \sin B = \frac{73.8 \cdot \sin 22.28^{\circ}}{25.4} \). However, the result \approx 1.099 is not valid for a sine function (which ranges from -1 to 1). This means the triangle as described cannot exist. Therefore, when solving triangles, validate each step to avoid such inconsistencies.

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