Let's solve the right triangle problem from your image.

Given:

• $C = 90^\circ$ (Right angle)
• $a = 19.1 \, \text{cm}$ (Opposite side)
• $c = 45.7 \, \text{cm}$ (Hypotenuse)
• Need to find $b$, $A$, and $B$.

Step 1: Find side $b$ using the Pythagorean theorem

$c^2 = a^2 + b^2$ $b = \sqrt{c^2 - a^2}$ Substitute the values: $b = \sqrt{45.7^2 - 19.1^2}$

Step 2: Find angle $A$ using the sine function

$\sin(A) = \frac{a}{c}$ $A = \sin^{-1}\left(\frac{19.1}{45.7}\right)$

Step 3: Find angle $B$ using the sum of angles in a triangle

$B = 90^\circ - A$

Let me calculate these for you.### Solutions:

• $b \approx 41.52 \, \text{cm}$ (rounded to the nearest hundredth)
• $A \approx 24^\circ 42'$ (rounded to the nearest minute)
• $B \approx 65^\circ 17'$ (rounded to the nearest minute)

Would you like more details or have any questions?

Here are 5 related questions to extend your learning:

1. How would the calculation change if side $a$ were doubled?
2. Can you solve the same triangle using cosine instead of sine?
3. What is the relationship between angles $A$ and $B$ in any right triangle?
4. How does the accuracy of your answer change if the values for $a$ or $c$ were given with more decimal places?
5. What are some common errors when applying the Pythagorean theorem?

Tip: Always double-check the accuracy of your angles by verifying that the sum of all angles in a triangle equals 180°.