If \(\cos θ = \frac 5 {13},\) then the value of tan 2 θ + sec 2 θ is equal to:
Evaluating Trigonometric Expression Value
This problem asks for the value of a trigonometric expression given the cosine of an angle \(\theta\). We are provided with:
\(\cos \theta = \frac{5}{13}\)
The expression to evaluate is \(\tan 2\theta + \sec 2\theta\). A direct calculation using standard double-angle formulas leads to a result of \(-17/7\). However, this result is negative, while the options provided are positive fractions with a denominator of 25. This significant discrepancy suggests that the question might have intended to ask for the value of \(\tan^2 \theta + \sec^2 \theta\). We will proceed with calculating \(\tan^2 \theta + \sec^2 \theta\) as this calculation aligns with the provided correct answer (Option 1).
Calculating tan2 θ + sec2 θ
Here is a step-by-step derivation:
Step 1: Determine the value of sec θ
The secant function (\(\sec \theta\)) is defined as the reciprocal of the cosine function (\(\cos \theta\)).
Formula: \(\sec \theta = \frac{1}{\cos \theta}\)
Substitute the given value of \(\cos \theta\):
\(\sec \theta = \frac{1}{5/13}\)
\(\sec \theta = \frac{13}{5}\)
Step 2: Compute sec2 θ
To find \(\sec^2 \theta\), we square the value of \(\sec \theta\):
\(\sec^2 \theta = \left(\frac{13}{5}\right)^2\)
\(\sec^2 \theta = \frac{13^2}{5^2} = \frac{169}{25}\)
Step 3: Compute tan2 θ
We utilize the Pythagorean trigonometric identity that relates \(\sec^2 \theta\) and \(\tan^2 \theta\):
Identity: \(\sec^2 \theta = 1 + \tan^2 \theta\)
Rearranging this identity to solve for \(\tan^2 \theta\):
\(\tan^2 \theta = \sec^2 \theta - 1\)
Substitute the computed value of \(\sec^2 \theta\):
\(\tan^2 \theta = \frac{169}{25} - 1\)
To subtract, we find a common denominator:
\(\tan^2 \theta = \frac{169}{25} - \frac{25}{25} = \frac{169 - 25}{25}\)
\(\tan^2 \theta = \frac{144}{25}\)
Step 4: Sum tan2 θ and sec2 θ
Finally, we add the computed values of \(\tan^2 \theta\) and \(\sec^2 \theta\) to find the value of the expression:
\(\tan^2 \theta + \sec^2 \theta = \frac{144}{25} + \frac{169}{25}\)
Since the denominators are the same, we simply add the numerators:
\(\tan^2 \theta + \sec^2 \theta = \frac{144 + 169}{25}\)
\(\tan^2 \theta + \sec^2 \theta = \frac{313}{25}\)
Final Answer Verification
The calculation shows that \(\tan^2 \theta + \sec^2 \theta = \frac{313}{25}\). This value matches Option 1 presented in the question.
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