Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on solving oblique triangles using the Law of Sines, including both the standard and ambiguous (SSA) cases. Key techniques involve determining unknown angles and sides using proportional relationships, computing the area of triangles using the sine area formula, and applying these methods to real-world problems. Understanding when to use the Law of Sines versus the Law of Cosines is crucial for accurately solving triangle problems.

Learning Objectives

1

Apply the Law of Sines to solve oblique triangles in AAS, ASA, and SSA cases.

2

Utilize the Law of Sines to determine unknown sides and angles, including the ambiguous SSA case.

3

Calculate the area of oblique triangles using the sine formula.

4

Model and solve real-life problems involving oblique triangles using trigonometric methods.

5

Contrast and combine the Law of Sines with the Law of Cosines for comprehensive triangle solutions.

Key Concepts

CONCEPT

DEFINITION

Oblique Triangle

A triangle that does not contain a right angle.

Law of Sines

A trigonometric formula that relates the ratios of the lengths of sides of a triangle to the sines of its opposite angles, expressed as a/sinA = b/sinB = c/sinC.

Ambiguous Case (SSA)

A situation in solving a triangle using two sides and a non-included angle where zero, one, or two triangles may satisfy the given conditions.

Area of an Oblique Triangle

The area can be determined by the formula Area = 1/2 * (side1) * (side2) * sin(included angle).

Law of Cosines

A trigonometric formula used to relate the sides and angles of a triangle when three sides (SSS) or two sides and their included angle (SAS) are known.

Example Problems

Example 1

An ________ triangle is a triangle that has no right angle.

Example 2

For triangle $ABC$, the Law of Sines is given by $\dfrac{a}{sin\ A}\ =$ ___________ $=\ \dfrac{c}{sin\ C}$.

Example 3

Two ________ and one ________ determine a unique triangle.

Example 4

The area of an oblique triangle is given by $\frac{1}{2}bc$ sin $A =\ \frac{1}{2}ab$ sin $C =$ ________ .

Example 5

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places.
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Step-by-Step Explanations

QUESTION

Given a triangle with angles A = 43°, B = 29° and side a = 28 ft, find the remaining angle and the lengths of sides b and c.

STEP-BY-STEP ANSWER:

Step 1: Find angle C using the triangle sum theorem: C = 180° - 43° - 29° = 108°.
Step 2: Apply the Law of Sines: a/sinA = b/sinB, hence b = (a * sinB) / sinA = (28 * sin29°) / sin43°.
Step 3: Calculate b using a calculator and round accordingly.
Step 4: Similarly, use a/sinA = c/sinC to find c = (28 * sin108°) / sin43°.
Step 5: Verify that the longest side is opposite the largest angle and check your sketch for consistency.
Final Answer: The triangle has angles 43°, 29°, 108° with sides approximately a = 28 ft, b = [calculated value] ft, and c = [calculated value] ft.

AAS Triangle (Oblique)

QUESTION

Given a triangle where side a = 22 in, side b = 12 in, and angle A = 42° are known, determine if one triangle, two triangles, or no triangle exists and solve for the unknown angles and side c if possible.

STEP-BY-STEP ANSWER:

Step 1: Compute the height h from side b using h = b * sin A = 12 * sin42°.
Step 2: Compare side a and the height h to determine feasibility: if a < h then no triangle; if a = h then one right triangle; if a > h then one or two triangles exist.
Step 3: Use the Law of Sines to solve for angle B: sinB = (b * sinA) / a.
Step 4: Notice that since sinB can yield two possible angles (one acute and one obtuse) if sinB < 1, there may be two solutions.
Step 5: Compute both possible values for B, then use angle C = 180° - A - B for each case.
Step 6: Finally, solve for side c using the relation c = (a * sinC) / sinA.
Final Answer: Depending on the computed values, either one unique triangle or two distinct triangles can be obtained.

Ambiguous Case (SSA)

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Common Mistakes

  • Failing to correctly identify which side is opposite which angle when applying the Law of Sines.
  • Ignoring the possibility of two solutions in the ambiguous (SSA) case.
  • Neglecting to check for the feasibility of a triangle (e.g., ensuring the longest side is opposite the largest angle).
  • Rounding intermediate results too early, which can lead to inaccuracies.