Brief Summary
This video explains the law of sines, which is used to solve oblique triangles (triangles without a right angle). It covers the different forms of the law of sines, how to apply it when given different information about a triangle (two angles and a side, or two sides and an angle opposite one of them), and the ambiguous case, where there might be zero, one, or two possible triangles that fit the given information.
- Law of Sines is applicable for oblique triangles.
- Ambiguous case can result in zero, one, or two possible triangle solutions.
- The video provides step-by-step examples.
Introduction to the Law of Sines
The law of sines is used to solve oblique triangles, which are triangles that do not contain a right angle. To solve a triangle, you need to know three parts of it, with at least one of them being a side. Sides are represented by lowercase letters (a, b, c), and angles are represented by uppercase letters (A, B, C), where each side is opposite its corresponding angle. The law of sines can be written in three parts: sine A / a = sine B / b = sine C / c. It can also be written in its reciprocal form: a / sine A = b / sine B = c / sine C.
Forms and Application of the Law of Sines
The law of sines can take several forms when cross-multiplying to solve for an unknown side or angle. If two angles and one side are given, first find the third angle by subtracting the sum of the two given angles from 180 degrees, then use the law of sines to find the other two sides. The law of sines can be used when you know one side and two angles, or two sides and an angle opposite one of those sides.
Example 1: Finding Sides A and C
Given triangle ABC with angle A = 43.8 degrees, angle C = 62.2 degrees, and side b = 15.8 inches, the goal is to find sides a and c. First, find angle B using the triangle sum theorem: 180 - (43.8 + 62.2) = 74 degrees. Then, use the law of sines to find side a: a / sin(43.8) = 15.8 / sin(74). Cross-multiply to get a = (15.8 * sin(43.8)) / sin(74) = 11.38 inches.
Example 2: Finding Sides B and C
To solve for side c, use the law of sines again: c / sin(62.2) = 15.8 / sin(74). Cross-multiply to get c = (sin(62.2) * 15.8) / sin(74) = 14.54 inches. Next example involves finding sides b and c when given two angles and one side. Given angle A = 45 degrees, angle B = 50 degrees, and side a = 8, first find angle C: 180 - (45 + 50) = 85 degrees.
Example 3: Finding Sides B and C with Two Angles and One Side
To find side b, use the law of sines: b / sin(50) = 8 / sin(45). Cross-multiply to get b = (sin(50) * 8) / sin(45) = 8.67. To find side c, use the law of sines again: c / sin(85) = 8 / sin(45). Cross-multiply to get c = (sin(85) * 8) / sin(45) = 11.27.
The Ambiguous Case: Introduction
The ambiguous case occurs when given two sides and an angle opposite one of them, which can result in zero, one, or two possible triangles. There are three possible situations: no triangle exists, exactly one triangle exists, or two distinct triangles exist.
Condition 1: Acute Angle A
If angle A is acute (between 0 and 90 degrees), there are several cases to consider. If a < b * sin(A), then sin(B) > 1, and there is no solution. If a = b * sin(A), then sin(B) = 1, meaning angle B is a right angle, and there is exactly one (right) triangle. If a > b * sin(A) and a < b, there are two possible triangles and two solutions. If a > b * sin(A) and a ≥ b, there is exactly one triangle and one solution.
Condition 2: Obtuse Angle A
If angle A is obtuse (between 90 and 180 degrees), there are two cases. If a ≤ b, there is no solution. If a > b, there is exactly one triangle and one solution.
Example 4: Applying the Ambiguous Case
Given a = 12 cm, b = 23 cm, and angle A = 34 degrees, find angle B using the law of sines: sin(B) / 23 = sin(34) / 12. This gives sin(B) = (sin(34) * 23) / 12 = 1.072. Since sin(B) > 1, there is no solution, meaning no triangle can be formed with the given measurements.
Example 5: Solving for B and C with One Solution
Given angle A = 73 degrees, a = 18, and b = 11, solve for angles B and C. Using the law of sines, sin(B) / 11 = sin(73) / 18, so sin(B) = (sin(73) * 11) / 18 = 0.5844. Angle B = arcsin(0.5844) = 35.76 degrees. Since A > B (18 > 11), there is one solution. Angle C = 180 - (73 + 35.76) = 71.24 degrees.
Finding Side C
To find side c, use the law of sines: c / sin(71.24) = 18 / sin(73). Cross-multiply to get c = (sin(71.24) * 18) / sin(73) = 17.82 cm.
Example 6: Ambiguous Case with Two Possible Solutions
Given b = 16 cm, angle B = 28 degrees, and c = 20 cm, determine the number of possible solutions. Compare b with c * sin(B): 16 > 20 * sin(28) (16 > 9.39). Since B is an acute angle and b > c * sin(B), this is an ambiguous case with two possible solutions.
Finding the First Solution
To find angle C, use the law of sines: sin(C) / 20 = sin(28) / 16, so sin(C) = (sin(28) * 20) / 16 = 0.5868. Angle C1 = arcsin(0.5868) = 35.93 degrees. Angle A = 180 - (28 + 35.93) = 116.07 degrees.
Finding Side A for the First Solution
To find side a, use the law of sines: a / sin(116.07) = 16 / sin(28). Cross-multiply to get a = (sin(116.07) * 16) / sin(28) = 30.61 cm.
Finding the Second Solution
For the second solution, use the supplementary angle of C1: C2 = 180 - 35.93 = 144.07 degrees. Angle A = 180 - (28 + 144.07) = 7.93 degrees.
Finding Side A for the Second Solution
To find side a, use the law of sines: a / sin(7.93) = 16 / sin(28). Cross-multiply to get a = (sin(7.93) * 16) / sin(28) = 4.70 cm. Therefore, there are two possible triangles with the given information.
