Question

if \\(\sin x = \frac{4}{5}\\), where \\(x\\) is in quadrant i, then find each trigonometric function (without finding the angle \\(x\\)). your answers should be exact values (no decimals).
example solution input: if the solution contains a square root, such as \\(\frac{\sqrt{7}}{3}\\), enter sqrt(7)/3.
answers.
a. \\(\sin 2x = \boxed{}\\)
b. \\(\cos 2x = \boxed{}\\)
c. \\(\tan 2x = \boxed{}\\)
\\(\bullet\\) a solution write-up is required for this problem.

Explanation:

Step1: Find \(\cos x\)

Given \(\sin x=\frac{4}{5}\) and \(x\) in quadrant I. Use \(\sin^2 x+\cos^2 x = 1\).
\(\cos^2 x=1-\sin^2 x=1 - (\frac{4}{5})^2=1-\frac{16}{25}=\frac{9}{25}\).
Since \(x\) is in quadrant I, \(\cos x=\frac{3}{5}\).

Step2: Calculate \(\sin 2x\)

Use double - angle formula \(\sin 2x = 2\sin x\cos x\).
Substitute \(\sin x=\frac{4}{5}\) and \(\cos x=\frac{3}{5}\):
\(\sin 2x=2\times\frac{4}{5}\times\frac{3}{5}=\frac{24}{25}\).

Step3: Calculate \(\cos 2x\)

Use double - angle formula \(\cos 2x=\cos^2 x-\sin^2 x\) (or \(1 - 2\sin^2 x\) or \(2\cos^2 x - 1\)).
Using \(\cos^2 x-\sin^2 x\): \(\cos 2x=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25}\).

Step4: Calculate \(\tan 2x\)

Use the formula \(\tan 2x=\frac{\sin 2x}{\cos 2x}\).
Substitute \(\sin 2x=\frac{24}{25}\) and \(\cos 2x = -\frac{7}{25}\):
\(\tan 2x=\frac{\frac{24}{25}}{-\frac{7}{25}}=-\frac{24}{7}\).

Answer:

s:
A. \(\sin 2x=\frac{24}{25}\)
B. \(\cos 2x = -\frac{7}{25}\)
C. \(\tan 2x=-\frac{24}{7}\)