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Answer :
Sure! Let's break down the expression step by step to understand how to solve it.
The expression given is:
[tex]\[ 625 \cdot \sec^2\left(\tan^{-1}\left(\frac{13}{25}\right)\right) \cdot 8\pi \][/tex]
Step 1: Understand the Inverse Tangent
- [tex]\(\tan^{-1}\left(\frac{13}{25}\right)\)[/tex] represents the angle whose tangent is [tex]\(\frac{13}{25}\)[/tex].
Step 2: Find the Secant Squared
- A trigonometric identity states that for an angle [tex]\(\theta\)[/tex],
[tex]\[ \sec^2(\theta) = 1 + \tan^2(\theta) \][/tex]
- Since we know [tex]\(\tan(\theta) = \frac{13}{25}\)[/tex], we use this to find [tex]\(\sec^2(\theta)\)[/tex]:
[tex]\[ \tan^2(\theta) = \left(\frac{13}{25}\right)^2 = \frac{169}{625} \][/tex]
[tex]\[ \sec^2(\theta) = 1 + \frac{169}{625} = \frac{625}{625} + \frac{169}{625} = \frac{794}{625} = 1.2704 \][/tex]
Step 3: Use the Given Values to Calculate the Expression
- Now, substitute the values back into the original expression:
[tex]\[ 625 \cdot 1.2704 \cdot 8\pi \][/tex]
- When you calculate the product:
- First, multiply [tex]\(625\)[/tex] and [tex]\(1.2704\)[/tex], which gives approximately: [tex]\(793.95\)[/tex]
- Then, multiply [tex]\(793.95\)[/tex] by [tex]\(8\pi\)[/tex] to get the final result.
This leads us to the numerical outcome:
[tex]\[ 19955.3965 \][/tex]
So, the answer to the original expression is approximately [tex]\(19955.40\)[/tex].
I hope this step-by-step explanation helps you understand how to approach and solve this kind of problem!
The expression given is:
[tex]\[ 625 \cdot \sec^2\left(\tan^{-1}\left(\frac{13}{25}\right)\right) \cdot 8\pi \][/tex]
Step 1: Understand the Inverse Tangent
- [tex]\(\tan^{-1}\left(\frac{13}{25}\right)\)[/tex] represents the angle whose tangent is [tex]\(\frac{13}{25}\)[/tex].
Step 2: Find the Secant Squared
- A trigonometric identity states that for an angle [tex]\(\theta\)[/tex],
[tex]\[ \sec^2(\theta) = 1 + \tan^2(\theta) \][/tex]
- Since we know [tex]\(\tan(\theta) = \frac{13}{25}\)[/tex], we use this to find [tex]\(\sec^2(\theta)\)[/tex]:
[tex]\[ \tan^2(\theta) = \left(\frac{13}{25}\right)^2 = \frac{169}{625} \][/tex]
[tex]\[ \sec^2(\theta) = 1 + \frac{169}{625} = \frac{625}{625} + \frac{169}{625} = \frac{794}{625} = 1.2704 \][/tex]
Step 3: Use the Given Values to Calculate the Expression
- Now, substitute the values back into the original expression:
[tex]\[ 625 \cdot 1.2704 \cdot 8\pi \][/tex]
- When you calculate the product:
- First, multiply [tex]\(625\)[/tex] and [tex]\(1.2704\)[/tex], which gives approximately: [tex]\(793.95\)[/tex]
- Then, multiply [tex]\(793.95\)[/tex] by [tex]\(8\pi\)[/tex] to get the final result.
This leads us to the numerical outcome:
[tex]\[ 19955.3965 \][/tex]
So, the answer to the original expression is approximately [tex]\(19955.40\)[/tex].
I hope this step-by-step explanation helps you understand how to approach and solve this kind of problem!
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