How do you get the second triangle when you have an ambiguous case of sine law? For example, if you are given the following information for a triangle:
a = 7.2 mm, b = 9.3 mm, <A = 35°
I can solve for one triangle:
Find <B, <C, side c.
<B:
sinB/9.3 = sin35°/72
sinB = 9.3sin35°/7.2
B = sin^-1 (9.3sin35°/7.2)
B = 48°
<C:
180°-35°-48° = 97°
Side c:
c/sin97° = 7.2/sin35°
c = 7.2/sin97°/sin35°
c = 12.5
How do you find and solve for the second one?
your calculator gave you 47.80.. or 48°
then angle C = 180-35-48 = 97 , you had that correct
c/sin97 = 7.2/sin35
c = 7.2sin97/sin35
= 12.5
So triangle ABC is such that
a= 7.2 --- given
b = 9.3 -- given
c = 12.5
Angle A = 35 -- given
Angle B = 48
Angle C = 97
but remember the sine is positive in quadrants I or II
so angle B could also have been 180-48 or 132 °
so c/sin48 = 7.2/sin35
c = 7.2sin48/sin35 = 9.3
so angle C = 180-132-35 = 13°
then c/sin13 = 7.2/sin35
c = 7.2sin13/sin35 = 2.8
second case
a=7.2
b=9.3
c= 2.8
A=35°
B = 132°
C = 13°
Ah, the ambiguous case of the sine law, it's like a puzzle with multiple solutions. So, to find the second triangle, you need to change your approach. Let's start with the given information:
a = 7.2 mm, b = 9.3 mm, <A = 35°
You've already solved for one triangle, now let's consider the second one. In this case, we'll begin by finding <c. Since the sum of angles in a triangle is 180°, we have:
<c = 180° - <A - <B
Plugging in the values:
<c = 180° - 35° - 48°
<c = 97°
Now we can use the sine law to find side a. We'll use the same ratio setup as before, but swap the values around:
sinA/a = sinC/c
Plugging in the values:
sin35°/a = sin97°/9.3
To solve for side a, we can cross-multiply and isolate it:
a = (9.3 * sin35°) / sin97°
a ≈ 4.417 mm
So, for the second triangle, we have:
a ≈ 4.417 mm, b = 9.3 mm, <A = 35°, <B = 48°, <C ≈ 97°
And there you have it! Two triangles for the price of one ambiguity. Just remember, in the world of math, sometimes there's more than one solution to a problem, just like in life.
To find the second triangle in an ambiguous case of the sine law, you need to consider the possibility of another triangle having the same given information.
In this case, you have found one triangle with angles A, B, and C as 35°, 48°, and 97° respectively, and side lengths a = 7.2 mm, b = 9.3 mm, and c ≈ 12.5 mm.
To find the second triangle, you need to consider the possibility of angle B being obtuse (greater than 90°). Let's assume that angle B in the second triangle is obtuse.
1. Find angle A (second triangle):
Angle A in the second triangle can be found by subtracting angle B from 180°:
A = 180° - 48°
A ≈ 132°
2. Find angle C (second triangle):
Angle C in the second triangle can be found by subtracting angles A and B from 180°:
C = 180° - 132° - 48°
C ≈ 0° (This doesn't seem to make sense, so this is not a valid triangle.)
Conclusion:
Based on the given information, it seems that there is only one valid triangle with angles 35°, 48°, and 97° and side lengths 7.2 mm, 9.3 mm, and approximately 12.5 mm. The assumption of an obtuse angle B does not lead to a valid triangle.
To find the second triangle in an ambiguous case of sine law, you need to consider two possibilities: one where angle B is acute (less than 90 degrees) and another where angle B is obtuse (greater than 90 degrees).
Using the information given (a = 7.2 mm, b = 9.3 mm, and <A = 35°), you have already solved for triangle ABC where angle B is acute.
To find the second triangle, where angle B is obtuse, you need to follow these steps:
Step 1: Find the possible range of values for angle B. Since angle B is obtuse, it must be greater than 90 degrees. Start by finding the supplement of angle B by subtracting its acute value (48 degrees) from 180 degrees. The supplement of angle B is 180 - 48 = 132 degrees.
Step 2: Find the possible range of values for side b. Use the sine law:
sin(B)/9.3 = sin(35°)/7.2
Rearrange the equation to solve for b:
b = (9.3 * sin(B))/sin(35°)
Now, you have a relationship between side b and angle B. Calculate the length of side b for two values of angle B: the acute value (48 degrees) and the supplement value (132 degrees).
For the acute triangle (triangle ABC), you already found b = 9.3 mm.
For the obtuse triangle (triangle A'B'C'), plug in the supplement value of angle B (132 degrees) into the equation above to find b':
b' = (9.3 * sin(132°))/sin(35°)
Step 3: Find angle C' for the obtuse triangle. Use the angle sum of a triangle:
angle C' = 180° - angle A - angle B'
angle C' = 180° - 35° - 132°
angle C' = 13°
Step 4: Calculate the length of side c' for the obtuse triangle. Use the sine law:
c'/sin(C') = 7.2/sin(35°)
c' = (7.2 * sin(13°))/sin(35°)
Now, you have found the lengths of sides b' and c' as well as angle C' for the obtuse triangle.
Therefore, the second triangle in the ambiguous case is triangle A'B'C' with side lengths a = 7.2 mm, b' (calculated), c' (calculated), and angles <A = 35°, <B' (supplement of 48°), <C' (calculated).