To solve the right triangle ABC, with $\angle C = 90^\circ$, $\angle A = 34.7^\circ$, and $c = 17.8 \, \text{ft}$ (where $c$ is the hypotenuse), we need to find:

• $\angle B$ (the other acute angle),
• side $a$ (the side opposite angle $A$),
• side $b$ (the side opposite angle $B$).

1. Find $\angle B$:

In a right triangle, the sum of the angles is always $180^\circ$. Since $\angle C = 90^\circ$ and $\angle A = 34.7^\circ$, we can find $\angle B$ using: $\angle B = 90^\circ - \angle A$ $\angle B = 90^\circ - 34.7^\circ = 55.3^\circ$

2. Find side $a$ (opposite $\angle A$):

We can use the sine function: $\sin(A) = \frac{a}{c}$ $a = c \cdot \sin(A)$ Substitute the given values: $a = 17.8 \cdot \sin(34.7^\circ)$ $a \approx 17.8 \cdot 0.5698 \approx 10.1 \, \text{ft}$

3. Find side $b$ (opposite $\angle B$):

We can use the cosine function: $\cos(A) = \frac{b}{c}$ $b = c \cdot \cos(A)$ Substitute the values: $b = 17.8 \cdot \cos(34.7^\circ)$ $b \approx 17.8 \cdot 0.8214 \approx 14.6 \, \text{ft}$

Final Answers:

• $\angle B = 55.3^\circ$
• $a = 10.1 \, \text{ft}$
• $b = 14.6 \, \text{ft}$

Would you like more details or further clarification?

Related Questions:

1. How can you use the tangent function to find side lengths in a right triangle?
2. What other methods can be used to solve right triangles?
3. How would the solution change if $\angle A$ was increased?
4. Can you find the area of this right triangle?
5. How would the side lengths change if the hypotenuse was increased?

Tip:

In any right triangle, once one acute angle is known, you can easily find the other since they always sum to $90^\circ$.