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Chapter 5: Problem 19

Determine the number of triangles with the given parts and solve each triangle. $$ \gamma=60^{\circ}, b=20, c=10 \sqrt{3} $$

Short Answer

Expert verified
One triangle exists with \( a = \sqrt{700 - 200 \sqrt{3}} \).

Step by step solution

01

- Identify Given Information

Given: \( \gamma = 60^{\circ} \), \( b = 20 \), and \( c = 10 \sqrt{3} \). \( \gamma \) is the angle between sides \( a \) and \( b \).
02

- Use Law of Cosines

Use the Law of Cosines to find side \( a \): \[ a^2 = b^2 + c^2 - 2bc\cos(\gamma) \] Substitute the given values: \[ a^2 = 20^2 + (10 \sqrt{3})^2 - 2 \cdot 20 \cdot 10 \sqrt{3} \cdot \cos(60^{\circ}) \].
03

- Simplify the Equation

Calculate each term: \[ a^2 = 400 + 300 - 2 \cdot 20 \cdot 10 \sqrt{3} \cdot \frac{1}{2} \] which simplifies to \[ a^2 = 400 + 300 - 200 \sqrt{3} \]
04

- Further Simplify

Combine like terms: \[ a^2 = 700 - 200 \sqrt{3} \]. Thus, \[ a = \sqrt{700 - 200 \sqrt{3}} \].
05

- Determine Number of Triangles

Since all given values and computations are valid, there is exactly one triangle that fits the given conditions.
06

- Solve Each Triangle Component

Since there is only one triangle, the sides and angles of the triangle have already been computed.

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Key Concepts

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

Law of Cosines
The Law of Cosines is a fundamental concept in trigonometry used to solve triangles. In any triangle, the Law of Cosines states: For any triangle with sides \(a\), \(b\), and \(c\) and an included angle \(纬\), the relationship can be written as: \[ a^2 = b^2 + c^2 - 2bc \cos(纬) \] This formula helps to find an unknown side when two sides and the included angle are known. It is especially useful when dealing with non-right-angled triangles. In our example, with \(纬 = 60^\text{circ}\), \(b = 20\), and \(c = 10 \sqrt{3}\), we can find \(a\) by substituting these values into the formula.
Triangle Properties
Triangles have several important properties. One of the key properties in a triangle is the sum of the angles, which always equals \(180^{\text{circ}}\). In any triangle:
  • The lengths of the sides are related to the angles between them.
  • The exterior angle of a triangle is equal to the sum of the two interior opposite angles.
  • The largest angle is always opposite the longest side.
For our specific triangle:
  • We know angle \(纬 = 60^{\text{circ}}\).
  • Side \(b\) is 20 units long.
  • Side \(c\) is \(10 \sqrt{3}\) units long.
Using these properties helps simplify finding other unknowns when applying trigonometric laws such as the Law of Cosines.
Trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. Here are some key concepts:
  • Sine, cosine, and tangent are primary trigonometric functions.
  • The Law of Sines states: \( \frac{a}{\text{sin}A} = \frac{b}{\text{sin}B} = \frac{c}{\text{sin}C} \).
  • The Law of Cosines, as discussed earlier, helps in finding unknown sides and angles in any triangle.
Trigonometry is essential in solving problems involving triangles, such as determining distances, angles, and side lengths. In our earlier problem, trigonometric principles help us solve for side \(a\) when given specific sides and an angle: \( \bbox[yellow]{a^2 = 20^2 + (10 \sqrt{3})^2 - 2 \cdot 20 \cdot 10 \sqrt{3} \cdot \text{cos}(60^{\text{circ}})} \) This illustrates how trigonometry is used for analytical problem solving.

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Most popular questions from this chapter

Find the acute angle between each pair of lines using the theorem on the angle between two vectors and the dot product. Round approximate answers to the nearest tenth of a degree. $$ y=2 x, y=\frac{1}{3} x $$

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