Question

cos theta = 4/5, 0˚< theta < 90˚ use the information given to find sin2theta, cos2theta, and tan 2theta

Answer 1

D edge

Explanation:

Answer 2

First calculate
`\sin\theta`.


`\sin^2\theta+\cos^2\theta=1\\\\\sin^2\theta=1-\cos^2\theta\\\\\sin^2\theta=1-\left((4)/(5)\right)^2\\\\\\\sin^2\theta=1-(16)/(25)\\\\\\\sin^2\theta=(9)/(25)\quad|√((\dots))\\\\\\\sin\theta=\sqrt{(9)/(25)}\\\\\\\boxed{\sin\theta=(3)/(5)}`

We take positive value of sinθ because 0° < θ < 90°
Now we could calculate sin2θ, cos2θ and tan2θ:


`\sin2\theta=2\sin\theta\cos\theta=2\cdot(3)/(5)\cdot(4)/(5)=(2\cdot3\cdot4)/(5\cdot5)=(24)/(25)\\\\\\\cos2\theta=\cos^2\theta-\sin^2\theta=\left((4)/(5)\right)^2-\left((3)/(5)\right)^2=(16)/(25)-(9)/(25)=(7)/(25)\\\\\\ \tan2\theta=(\sin2\theta)/(\cos2\theta)=((24)/(25))/((7)/(25))=(24\cdot25)/(7\cdot25)=(24)/(7)=3(3)/(7)`