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Suppose a = 11 and b = 4.
solve for sec(A)

To find \( \sec(A) \), we need to use the trigonometric definition of secant, specifically:

\[
\sec(A) = \frac{1}{\cos(A)}
\]

From the triangle, we can use the definition of cosine for angle \( A \) in terms of the side lengths:

\[
\cos(A) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{b}{c}
\]

### Step 1: Use the Pythagorean Theorem to Find \( c \)

Since \( a = 11 \) and \( b = 4 \), we can find \( c \) (the hypotenuse) using the Pythagorean theorem:

\[
c = \sqrt{a^2 + b^2} = \sqrt{11^2 + 4^2} = \sqrt{121 + 16} = \sqrt{137}
\]

So, \( c = \sqrt{137} \).

### Step 2: Find \( \cos(A) \)

Now, substitute \( b \) and \( c \) into the cosine formula:

\[
\cos(A) = \frac{b}{c} = \frac{4}{\sqrt{137}}
\]

### Step 3: Find \( \sec(A) \)

Now, \( \sec(A) \) is the reciprocal of \( \cos(A) \):

\[
\sec(A) = \frac{1}{\cos(A)} = \frac{1}{\frac{4}{\sqrt{137}}} = \frac{\sqrt{137}}{4}
\]

### Final Answer

\[
\sec(A) = \frac{\sqrt{137}}{4}
\]

Would you like further details on any step? 

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### Related Questions:
1. How do we find \(\tan(A)\) given \(a\) and \(b\)?
2. What is the value of \(\sec(B)\) in this triangle?
3. How would the solution change if \(a\) and \(b\) were different values?
4. Can we use trigonometric identities to verify \(\sec(A)\) in a different way?
5. How do we find \(\sin(A)\) in this triangle?

### Tip:
For right triangles, remember that trigonometric ratios like \(\sin\), \(\cos\), and \(\tan\) can be directly calculated from the side lengths, helping you solve for other trigonometric functions easily.

Suppose a = 11 and b = 4. Solve for sec(A)

Learn how to solve for sec(A) in a right triangle where a = 11 and b = 4. This solution uses trigonometric identities and the Pythagorean theorem to find the secant of angle A. Suitable for high school students learning trigonometry.

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