Solve the right triangle​ ABC, with Cequals90degrees.
Aequals36degrees 15prime​, cequals414.5 ft


To solve the right triangle ABC where:

- \( C = 90^\circ \),
- \( A = 36^\circ 15' \),
- \( c = 414.5 \, \text{ft} \) (hypotenuse),

we need to find the other angles and the other two sides: \( B \), \( a \), and \( b \).

### Step 1: Find angle \( B \)
Since the sum of the angles in a triangle is 180°, and we know that angle \( C = 90^\circ \) and angle \( A = 36^\circ 15' \), angle \( B \) can be calculated as:
\[
B = 90^\circ - A = 90^\circ - 36^\circ 15'
\]
First, convert \( 36^\circ 15' \) to decimal degrees:
\[
36^\circ 15' = 36^\circ + \frac{15}{60} = 36.25^\circ
\]
Now subtract:
\[
B = 90^\circ - 36.25^\circ = 53.75^\circ
\]
So, \( B = 53^\circ 45' \).

### Step 2: Find side \( a \) (opposite angle \( A \))
Using the sine function:
\[
\sin(A) = \frac{a}{c}
\]
\[
a = c \cdot \sin(A) = 414.5 \cdot \sin(36.25^\circ)
\]

### Step 3: Find side \( b \) (opposite angle \( B \))
Using the cosine function:
\[
\cos(A) = \frac{b}{c}
\]
\[
b = c \cdot \cos(A) = 414.5 \cdot \cos(36.25^\circ)
\]

Now let's calculate the values.The sides of the right triangle ABC are:

- \( a \approx 245.10 \, \text{ft} \) (opposite angle \( A \)),
- \( b \approx 334.27 \, \text{ft} \) (opposite angle \( B \)).

Thus, the triangle is fully solved with:

- \( A = 36^\circ 15' \),
- \( B = 53^\circ 45' \),
- \( c = 414.5 \, \text{ft} \),
- \( a \approx 245.10 \, \text{ft} \),
- \( b \approx 334.27 \, \text{ft} \).

Would you like any further details or clarification?

Here are 5 related questions to expand on this:
1. How do you convert angles from degrees and minutes to decimal form?
2. What is the significance of the sine and cosine functions in right triangles?
3. How can you use the tangent function to solve for sides in a right triangle?
4. What are some real-world applications of solving right triangles?
5. How would the calculations change if you were given the side lengths instead of angles?

**Tip:** Always make sure your calculator is in the correct mode (degrees/radians) when working with trigonometric functions.

Solve the right triangle ABC, with C = 90 degrees, A = 36 degrees 15', and c = 414.5 ft.

Learn how to solve a right triangle with angle C = 90°, angle A = 36°15', and hypotenuse c = 414.5 ft. This step-by-step solution covers key trigonometric concepts, including the sine and cosine functions, and angle-side relationships. Suitable for students in Grades 9-12.

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