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Use the tangent ratio to solve the missing side length AB of the right angle. Round to the nearest hundredths. C=26 degrees. BC=15 inches

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To find the missing side length \( AB \) of the right triangle using the tangent ratio, we can use the definition of the tangent function:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In this triangle, angle \( C \) is \( 26^\circ \), \( BC \) (the side opposite angle \( C \)) is given as \( 15 \) inches, and we need to find the length of side \( AB \) (the side adjacent to angle \( C \)).

Here, we have:

  • Opposite side (BC) = \( 15 \) inches
  • Angle \( C \) = \( 26^\circ \)
  • Adjacent side (AB) = ?

Using the tangent function:

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

We can rearrange this to find \( AB \):

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

Now, substitute \( BC = 15 \) inches into the equation:

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

Using a calculator to find \( \tan(26^\circ) \):

\[ \tan(26^\circ) \approx 0.4877 \]

Now, substitute this value back into the equation:

\[ AB = \frac{15}{0.4877} \approx 30.73 \text{ inches} \]

Therefore, the length of side \( AB \) rounded to the nearest hundredths is:

\[ \boxed{30.73} \text{ inches} \]

So the answer is 30.73 inches?

Yes, that's correct! The length of side \( AB \) is approximately \( 30.73 \) inches when rounded to the nearest hundredths place. If you have any further questions or need additional clarification, feel free to ask!

That was incorrect

I apologize for the confusion. Let's go through the calculations step-by-step again to ensure we get the correct result.

Given Information:

  • Angle \( C = 26^\circ \)
  • Opposite side \( BC = 15 \) inches

Step Using Tangent Ratio:

\[ \tan(C) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} \]

Rearranging gives us: \[ AB = \frac{BC}{\tan(C)} \]

Substituting the values we have: \[ AB = \frac{15}{\tan(26^\circ)} \]

Calculating \(\tan(26^\circ)\):

Using a calculator: \[ \tan(26^\circ) \approx 0.4877 \]

Substituting back into the equation:

\[ AB = \frac{15}{0.4877} \approx 30.73 \text{ inches} \]

Conclusion

The value of \( AB \) was indeed calculated as approximately \( 30.73 \) inches. If this does not match your expected answer, please provide additional context or details about the problem, such as confirming the triangle configuration or any side labels, so I can help accurately. Thank you!