Find $\sin 2 x, \cos 2 x$ , and $\tan 2 x$ if $\cos x=\frac{3}{5}$ and $x$ terminates in quadrant $I$ .

View step-by-step help

Home

Math Problems

Algebra 2

Quadrants

Full solution

Q. Find

\sin 2 x, \cos 2 x

, and

\tan 2 x

\cos x=\frac{3}{5}

and

x

terminates in quadrant

I

Find $\sin x$ : Use the given information to find $\sin x$ . $\newline$ Since $\cos x = \frac{3}{5}$ and $x$ is in quadrant I, where all trigonometric functions are positive, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to find $\sin x$ . $\newline$ Let's calculate $\sin x$ : $\newline$ $\sin^2 x = 1 - \cos^2 x$ $\newline$ $\sin^2 x = 1 - \left(\frac{3}{5}\right)^2$ $\newline$ $\sin^2 x = 1 - \frac{9}{25}$ $\newline$ $\sin x$ $0$ $\newline$ $\sin x$ $1$ $\newline$ $\sin x$ $2$ $\newline$ $\sin x$ $3$
Find $\sin 2x$ , $\cos 2x$ , $\tan 2x$ : Use the double angle formulas to find $\sin 2x$ , $\cos 2x$ , and $\tan 2x$ . The double angle formulas are: $\sin 2x = 2 \sin x \cos x$ $\cos 2x = \cos^2 x - \sin^2 x$ $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$ We already know $\sin x$ and $\cos 2x$ $0$ , so we can plug these values into the formulas. Let's calculate $\sin 2x$ : $\cos 2x$ $2$ $\cos 2x$ $3$ $\cos 2x$ $4$
Calculate $\cos 2x$ : Calculate $\cos 2x$ using the double angle formula. $\newline$ $\cos 2x = \cos^2 x - \sin^2 x$ $\newline$ $\cos 2x = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2$ $\newline$ $\cos 2x = \frac{9}{25} - \frac{16}{25}$ $\newline$ $\cos 2x = -\frac{7}{25}$
Calculate $\tan 2x$ : Calculate $\tan 2x$ using the double angle formula. $\newline$ First, we need to find $\tan x$ , which is $\sin x / \cos x$ . $\newline$ $\tan x = \sin x / \cos x$ $\newline$ $\tan x = (4/5) / (3/5)$ $\newline$ $\tan x = 4/3$ $\newline$ Now, we can use this to find $\tan 2x$ : $\newline$ $\tan 2x = (2 \cdot \tan x) / (1 - \tan^2 x)$ $\newline$ $\tan 2x = (2 \cdot (4/3)) / (1 - (4/3)^2)$ $\newline$ $\tan 2x$ $0$ $\newline$ $\tan 2x$ $1$ $\newline$ $\tan 2x$ $2$ $\newline$ $\tan 2x$ $3$ $\newline$ $\tan 2x$ $4$ $\newline$ $\tan 2x$ $5$