Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

The Law of Sines is a powerful tool for solving oblique triangles, whether given two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA). It not only enables the computation of unknown sides and angles but also plays a crucial role in calculating the area of a triangle. A critical aspect to master is the ambiguous case in SSA, where careful consideration is needed to decide if no triangle, one triangle, or two distinct triangles satisfy given conditions. These skills are widely applicable in solving real-world problems in navigation, architecture, and physics.

Learning Objectives

1

Apply the Law of Sines to solve oblique triangles given two angles and one side (AAS or ASA) and two sides and one angle (SSA).

2

Determine the conditions under which a triangle has one solution, no solution, or two solutions using the ambiguous case (SSA).

3

Calculate the area of oblique triangles using the formula Area = 1/2 bc sin A and its variants.

4

Model real-life situations, such as navigation and engineering problems, using the Law of Sines.

Key Concepts

CONCEPT

DEFINITION

Law of Sines

A trigonometric relation in any triangle given by a/sin A = b/sin B = c/sin C that is used to solve for unknown sides or angles.

Oblique Triangle

A triangle that does not contain a right angle; solved using the Law of Sines (or Law of Cosines in other cases).

Ambiguous Case (SSA)

A scenario in solving triangles where two sides and an angle not between them are given, which can lead to zero, one, or two possible triangles.

AAS/ASA

Types of triangle configurations where two angles and one side (AAS or ASA) are known, generally yielding a unique solution.

Area Formula for an Oblique Triangle

The area can be computed by Area = ½ ab sin C, or equivalently ½ bc sin A or ½ ac sin B, where the sine of the included angle is used.

Example Problems

Example 1

Fill in the blank(s). A(n) _______ triangle has no right angles.

Example 2

Fill in the blank(s). Law of sines: $\frac{a}{\sin A}=$ ______________$=\frac{c}{\sin C}$

Example 3

Fill in the blank(s). To find the area of any triangle, use one of the three formulas: Area $=$ _____________, ___________or ___________.

Example 4

Fill in the blank(s). Two___________ and one_____________determine a unique triangle.

Example 5

Which two cases can be solved using the Law of Sines?
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Step-by-Step Explanations

QUESTION

Given a triangle with angles C = 102.3°, B = 28.7° and side b = 27.4 ft, find angle A and sides a and c.

STEP-BY-STEP ANSWER:

Step 1: Find the third angle using A = 180° - B - C = 180° - 28.7° - 102.3° = 49.0°.
Step 2: Apply the Law of Sines: a/sin A = b/sin B = c/sin C.
Step 3: Solve for a: a = b * (sin A / sin B) = 27.4 * (sin 49.0° / sin 28.7°) ≈ 43.06 ft.
Step 4: Solve for c: c = b * (sin C / sin B) = 27.4 * (sin 102.3° / sin 28.7°) ≈ 55.75 ft.
Final Answer: The triangle has angles A = 49.0°, B = 28.7°, C = 102.3° and sides a ≈ 43.06 ft, b = 27.4 ft, c ≈ 55.75 ft.

Solving an AAS Triangle

QUESTION

For a triangle with a = 12 m, b = 31 m, and angle A = 20.5°, find the two possible triangles.

STEP-BY-STEP ANSWER:

Step 1: Compute the height h = b sin A = 31 * sin(20.5°) ≈ 10.86 m.
Step 2: Since h < a < b, two solutions exist.
Step 3: Use the Law of Sines to find sin B: sin B = b * (sin A / a) = 31 * (sin 20.5° / 12) ≈ 0.9047.
Step 4: Find the two possible values for angle B: B₁ ≈ sin⁻¹(0.9047) ≈ 64.8° and B₂ = 180° - 64.8° ≈ 115.2°.
Step 5: Calculate angle C for each case: For B₁, C = 180° - 20.5° - 64.8° = 94.7°; for B₂, C = 180° - 20.5° - 115.2° = 44.3°.
Step 6: Find side c using c = a * (sin C / sin A): For case 1, c ≈ 12 * (sin 94.7° / sin 20.5°) ≈ 34.15 m; For case 2, c ≈ 12 * (sin 44.3° / sin 20.5°) ≈ 23.93 m.
Final Answer: Two triangle solutions exist with (B, C, c) approximately (64.8°, 94.7°, 34.15 m) and (115.2°, 44.3°, 23.93 m).

Handling the Ambiguous Case (SSA)

QUESTION

Find the area of a triangular lot with sides 90 m and 52 m and an included angle of 102°.

STEP-BY-STEP ANSWER:

Step 1: Identify the given side lengths a = 90 m, b = 52 m, and included angle C = 102°.
Step 2: Use the area formula: Area = ½ ab sin C.
Step 3: Substitute the given values: Area = ½ * 90 * 52 * sin(102°).
Step 4: Compute sin 102° and finalize the calculation: Area ≈ ½ * 4680 * sin(102°) ≈ 2288.87 square meters.
Final Answer: The area of the triangle is approximately 2288.87 m².

Finding the Area of an Oblique Triangle

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Common Mistakes

  • Failing to ensure that the computed sine values are within the valid range (-1 to 1), which may lead to false conclusions about the existence of a triangle.
  • Ignoring the ambiguous case in SSA scenarios; not checking for the possibility of having two solutions (or no solution) based on the relation between side lengths and the altitude.
  • Misidentifying the largest side or angle, which can lead to inconsistent or infeasible triangle sketches.
  • Using the wrong formula form by not isolating the unknown properly in the numerator, which may cause calculation errors.