Question

question
given \\( \tan a = \frac { 4 } { 5 } \\) and that angle \\( a \\) is in quadrant i, find the exact value of \\( \cos a \\) in simplest radical form using a rational denominator.
answer attempt 1 out of 2
submit answer

Explanation:

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

Given \(\tan A=\frac{4}{5}\), then \(\tan^{2}A = (\frac{4}{5})^{2}=\frac{16}{25}\).
Substitute into the identity: \(1+\frac{16}{25}=\sec^{2}A\).
Calculate \(1+\frac{16}{25}=\frac{25 + 16}{25}=\frac{41}{25}\), so \(\sec^{2}A=\frac{41}{25}\).

Step2: Find \(\cos A\)

Since \(\sec A=\frac{1}{\cos A}\) and \(\sec^{2}A=\frac{41}{25}\), then \(\cos^{2}A=\frac{25}{41}\).
Because angle \(A\) is in Quadrant I (\(\cos A>0\) in Quadrant I), \(\cos A=\sqrt{\frac{25}{41}}=\frac{5}{\sqrt{41}}\).
Rationalize the denominator: \(\cos A=\frac{5\sqrt{41}}{41}\).

Answer:

\(\frac{5\sqrt{41}}{41}\)