. In the triangle ABC, c = 17.35, a = 26.82 and A = 101°47′
. Calculate C, B and b. how do i get 39°18′
, 38°55′
, 17.21


To solve the triangle \(ABC\) given \(c = 17.35\), \(a = 26.82\), and \(\angle A = 101^\circ 47'\), you need to determine the angles \(B\) and \(C\) and the side \(b\). Here’s the step-by-step solution:

### 1. **Convert the given angle A to decimal degrees**:
\[
\angle A = 101^\circ 47' = 101^\circ + \frac{47}{60}^\circ = 101.7833^\circ
\]

### 2. **Use the Law of Sines**:
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Start by finding \(\sin A\):
\[
\sin A = \sin 101.7833^\circ
\]

Now use the Law of Sines to find \(\sin C\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} \Rightarrow \sin C = \frac{c \cdot \sin A}{a}
\]
Substitute the values \(c = 17.35\), \(a = 26.82\), and \(\sin A\) (calculated earlier):
\[
\sin C = \frac{17.35 \cdot \sin(101.7833^\circ)}{26.82}
\]

### 3. **Calculate \(\angle C\)**:
\[
\angle C = \arcsin\left(\sin C\right)
\]

Given that the sum of angles in a triangle is \(180^\circ\):
\[
\angle B = 180^\circ - \angle A - \angle C
\]

### 4. **Calculate side \(b\)** using the Law of Sines**:
\[
\frac{b}{\sin B} = \frac{a}{\sin A} \Rightarrow b = \frac{a \cdot \sin B}{\sin A}
\]

After performing these calculations:

- \(\angle C = 39^\circ 18'\)
- \(\angle B = 38^\circ 55'\)
- \(b = 17.21\)

### Steps for These Results:

1. **Angle C Calculation**: Convert \(39^\circ 18'\) to decimal degrees and find its sine to cross-check.
2. **Angle B Calculation**: Calculate by subtracting \(\angle A\) and \(\angle C\) from 180°.
3. **Side b Calculation**: Use \(\sin B\) found from the previous step in the Law of Sines to calculate \(b\).

Would you like a more detailed breakdown of any specific calculation steps? 

Here are 5 questions that build on this topic:

1. How is the Law of Sines derived from the properties of triangles?
2. What are the limitations of the Law of Sines in solving triangles?
3. How do you handle ambiguous cases (SSA) when using the Law of Sines?
4. How do you convert between degrees, minutes, and seconds to decimal degrees?
5. How do you verify the results of triangle calculations using different methods?

**Tip:** Always ensure angles and sides are consistently in the same units when applying trigonometric laws.

Learn how to solve triangle ABC with given side c = 17.35, side a = 26.82, and angle A = 101°47′. Find angles C and B, and side b using the Law of Sines and angle sum theorem. Detailed step-by-step solution provided.

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