Question

question
given tan a = 7/4 and that angle a is in quadrant i, find the exact value of sec a in simplest radical form using a rational denominator.
answer
attempt 1 out of 2

Explanation:

Step1: Use the identity \(1 + \tan^{2}A=\sec^{2}A\)

Given \(\tan A=\frac{7}{4}\), then \(\tan^{2}A = (\frac{7}{4})^{2}=\frac{49}{16}\). Substituting into the identity: \(1+\frac{49}{16}=\sec^{2}A\).

Step2: Calculate the left - hand side

\(1+\frac{49}{16}=\frac{16 + 49}{16}=\frac{65}{16}\), so \(\sec^{2}A=\frac{65}{16}\).

Step3: Determine the sign of \(\sec A\)

Since angle \(A\) is in Quadrant I, \(\sec A>0\). Then \(\sec A=\sqrt{\frac{65}{16}}=\frac{\sqrt{65}}{4}\)

Answer:

\(\frac{\sqrt{65}}{4}\)