Question
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
Answers
GPT-4o mini
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:
\[
\boxed{34.19} \text{ inches}
\]
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:
\[
\boxed{34.19} \text{ inches}
\]
K
That wasn’t correct
GPT-4o mini
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!
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!