Solve the right triangle​ ABC, with Cequals90degrees.
Bequals34degrees10prime​, cequals0.6222 m
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Part 1
Aequals
  
enter your response heredegrees 
  
enter your response hereprime ​(Simplify your answers. Type integers or​ decimals.)
aequals
  
enter your response here m ​(Simplify your answer. Type an integer or a decimal. Round to four decimal places as​ needed.)
bequals
  
enter your response here m ​(Simplify your answer. Type an integer or a decimal. Round to four decimal places as​ needed.)

To solve the right triangle \( \triangle ABC \), we are given the following information:

- \( C = 90^\circ \) (right angle)
- \( B = 34^\circ 10' \) (angle B)
- \( c = 0.6222 \, \text{m} \) (the hypotenuse)

We need to find:

- \( A \) (angle A)
- \( a \) (side opposite angle A)
- \( b \) (side opposite angle B)

### Step 1: Find angle \( A \)

Since the sum of the angles in any triangle is \( 180^\circ \), we can find angle \( A \) as:

\[
A = 180^\circ - 90^\circ - B
\]

Convert \( B \) to decimal form:
\[
B = 34^\circ 10' = 34 + \frac{10}{60} = 34.1667^\circ
\]

Now calculate \( A \):

\[
A = 180^\circ - 90^\circ - 34.1667^\circ = 55.8333^\circ
\]

Convert \( A \) back to degrees and minutes:

\[
A = 55^\circ 50'
\]

### Step 2: Find side \( a \)

We can use the sine of angle \( A \) to find side \( a \), since \( \sin(A) = \frac{a}{c} \).

\[
a = c \times \sin(A)
\]

First, calculate \( \sin(A) \):

\[
\sin(55.8333^\circ) \approx 0.8269
\]

Now calculate \( a \):

\[
a = 0.6222 \times 0.8269 \approx 0.5147 \, \text{m}
\]

### Step 3: Find side \( b \)

We can use the sine of angle \( B \) to find side \( b \), since \( \sin(B) = \frac{b}{c} \).

\[
b = c \times \sin(B)
\]

First, calculate \( \sin(34.1667^\circ) \):

\[
\sin(34.1667^\circ) \approx 0.5612
\]

Now calculate \( b \):

\[
b = 0.6222 \times 0.5612 \approx 0.3491 \, \text{m}
\]

### Final Answer:

- \( A = 55^\circ 50' \)
- \( a = 0.5147 \, \text{m} \)
- \( b = 0.3491 \, \text{m} \)

Would you like more details or have any questions?

Here are five related questions:
1. How would you solve the triangle if only one side and one angle were given?
2. Can you calculate the area of this triangle?
3. What trigonometric ratios are used to solve right triangles?
4. How do you convert between degrees and minutes in angle measurements?
5. How does the Pythagorean theorem apply to right triangles?

**Tip:** For right triangles, knowing one acute angle and one side is often enough to determine the entire triangle using trigonometric functions.

Solve the right triangle ABC, with C = 90 degrees, B = 34 degrees 10', c = 0.6222 m. Find angle A, side a, and side b.

This detailed solution explains how to solve a right triangle ABC, where C = 90°, B = 34°10', and hypotenuse c = 0.6222 m. Learn how to find angle A, side a, and side b using trigonometric ratios and the angle sum property. Suitable for students in Grades 9-11.

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