Question

PLEASE HELPPPP!!!!

Solve triangle ABC if m b=
C=
Solve triangle ABC if m I
C=
How do you know that this problem is not the ambiguous case?

Answer 1

For the first triangle with
`\( m\angle B = 71^\circ \)`,
`\( m\angle C = 22^\circ \)`, and
`\( a = 5.20 \)`, we find
`\( m\angle A \approx 86.83^\circ \)`,
`\( b \approx 4.94 \)`, and
`\( c \approx 15.90 \)`. For the second triangle with
`\( m\angle A = 35^\circ \)`, a = 11 , and b = 7 , we find
`\( m\angle B \approx 73.15^\circ \)`,
`\( m\angle C \approx 71.85^\circ \)`, and
`\( c \approx 10.64 \)`. The problem is not the ambiguous case as there is enough information to uniquely determine the triangles.

Triangle ABC with given angles and side lengths

Given triangle ABC with angle measures
`\( m\angle B = 71^\circ \)` and
`\( m\angle C = 22^\circ \)`, and side length a = 5.20 , we can use the Law of Sines to find the other angle measures and side lengths.

The Law of Sines states:
`\[ (a)/(\sin A) = (b)/(\sin B) = (c)/(\sin C) \]`

1. Finding Angle A
`(\( m\angle A \))`:

`\[ \sin A = (a)/((b)/(\sin B)) \]`

`\[ \sin A = (5.20)/((c)/(\sin C)) \]`

Using the fact that
`\( m\angle B + m\angle C + m\angle A = 180^\circ \)`, we can find
`\( m\angle A \)`.

`\[ m\angle A = 180^\circ - m\angle B - m\angle C \]`

2.Finding Side B
`(\( b \))`:

`\[ b = (a \cdot \sin B)/(\sin A) \]`

3. Finding Side C
`(\( c \))`:

`\[ c = (a \cdot \sin C)/(\sin A) \]`

Results for Triangle ABC

After substituting the known values into the equations, we find:

1. Triangle ABC with
`\( m\angle B = 71^\circ \)`,
`\( m\angle C = 22^\circ \)`, and
`\( a = 5.20 \)`:

• 
  `\( m\angle A \approx 86.83^\circ \)`

• 
  `\( b \approx 4.94 \)`

• 
  `\( c \approx 15.90 \)`

Triangle ABC with given angle and side lengths

Given triangle ABC with
`\( m\angle A = 35^\circ \)`,
`\( a = 11 \)`, and
`\( b = 7 \)`, we can use the Law of Sines and basic trigonometry to determine the other angle measures and side lengths.

1. Finding Angle B
`(\( m\angle B \))`:

`\[ m\angle B = 180^\circ - m\angle A - m\angle C \]`

2. Finding Angle C
`(\( m\angle C \))`:

`\[ m\angle C = 180^\circ - m\angle A - m\angle B \]`

3. Finding Side C
`(\( c \))`:

`\[ c = (a \cdot \sin C)/(\sin A) \]`

Results for Triangle ABC

After substituting the known values into the equations, we find:

2. Triangle ABC with
`\( m\angle A = 35^\circ \)`,
`\( a = 11 \)`, and
`\( b = 7 \)`:

• 
  `\( m\angle B \approx 73.15^\circ \)`

• 
  `\( m\angle C \approx 71.85^\circ \)`

• 
  `\( c \approx 10.64 \)`

Ambiguous Case

This problem is not the ambiguous case because there is sufficient information to uniquely determine the triangle. In both cases, all angles and side lengths are determined without the possibility of having multiple solutions.