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What is the side length of AB of the right triangle if C is 26 degrees and BC is 15 inches and B is 90 degrees? Round to the nearest hundredth

3 answers

To find the length of side AB in the right triangle where angle B is 90 degrees, angle C is 26 degrees, and side BC (the side opposite angle C) is 15 inches, we can use trigonometric ratios.

In a right triangle:

  • We can use the sine function which is defined as:

    \[ \sin(C) = \frac{\text{opposite}}{\text{hypotenuse}} \]

In this case, the side opposite to angle C (BC) is 15 inches, and we need to find side AB, which is the hypotenuse.

Let \( AB \) be the hypotenuse. Then:

\[ \sin(26^\circ) = \frac{BC}{AB} = \frac{15}{AB} \]

Rearranging to solve for \( AB \):

\[ AB = \frac{15}{\sin(26^\circ)} \]

Now, we need to calculate \( \sin(26^\circ) \). Using a calculator:

\[ \sin(26^\circ) \approx 0.4384 \]

Now we can substitute this value to find \( AB \):

\[ AB \approx \frac{15}{0.4384} \approx 34.19 \text{ inches} \]

Therefore, the length of side AB, rounded to the nearest hundredth, is:

\[ \boxed{34.19} \text{ inches} \]

That wasn’t correct

I apologize for the mistake. Let's go through the calculation step-by-step again to find the correct length of side AB.

Given:

  • Angle C = 26 degrees
  • Angle B = 90 degrees (implying this is a right triangle)
  • Side BC (the side opposite angle C) = 15 inches

To find AB, we will use the cosine function since we want the adjacent side to angle C (AB):

\[ \cos(C) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

In this case:

\(\cos(26^\circ) = \frac{AB}{BC}\)

Rearranging gives us:

\[ AB = BC \cdot \cos(26^\circ) \]

Now, substitute the values we know. You also need to know that BC = 15 inches:

\[ AB = 15 \cdot \cos(26^\circ) \]

Now, calculating \(\cos(26^\circ)\) using a calculator:

\[ \cos(26^\circ) \approx 0.8988 \]

Now we substitute this back in:

\[ AB \approx 15 \cdot 0.8988 \approx 13.48 \text{ inches} \]

So, the length of side AB, rounded to the nearest hundredth, is:

\[ \boxed{13.48} \text{ inches} \]

Thank you for your patience, and I appreciate the opportunity to correct my previous answer!