Question

Help me please i need this to be answered quickly!!!!!!

Answer 1

`\tan 2 \theta=-(120)/(119)`

Explanation:

In quadrant I of the Cartesian coordinate system, both the x-coordinate (cosine) and the y-coordinate (sine) of a point on the unit circle are positive. This means that if an angle θ is in quadrant I, then both sin θ and cos θ are positive.

Given sin θ = 12/13, we can calculate cos θ by using the trigonometric identity sin²θ + cos²θ = 1:

`\begin{aligned}\sin^2\theta+\cos^2\theta&=1\\\\\cos^2\theta&=1-\sin^2\theta\\\\\cos\theta&=√(1-\sin^2\theta)\\\\\cos\theta&=\sqrt{1-\left((12)/(13)\right)^2}\\\\\cos\theta&=(5)/(13)\end{aligned}`

Since the tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ), and both sin θ and cos θ are positive in quadrant I, then tan θ is also positive in quadrant I.

Calculate tan θ by using the trigonometric ratio tan θ = sin θ / cos θ:

`\begin{aligned}\tan \theta&=(\sin \theta)/(\cos \theta)\\\\\tan \theta&=((12)/(13))/((5)/(13))\\\\\tan \theta&=(12)/(5)\end{aligned}`

Finally, calculate tan 2θ using the Tangent Double Angle formula:

`\begin{aligned}\tan2\theta&=(2\tan\theta)/(1-\tan^2\theta)\\\\\tan2\theta&=(2 \cdot (12)/(5))/(1-\left((12)/(5)\right)^2)\\\\\tan2\theta&=((24)/(5))/(1-(144)/(25))\\\\\tan2 \theta&=((24)/(5))/((25)/(25)-(144)/(25))\\\\\tan2\theta&=((24)/(5))/((-119)/(25))\\\\\tan2\theta&=((24\cdot 5)/(5\cdot 5))/((-119)/(25))\\\\\tan2\theta&=((120)/(25))/((-119)/(25))\\\\\tan2\theta&=-(120)/(119)\end{aligned}`

Therefore:

`\large\boxed{\boxed{\tan 2 \theta=-(120)/(119)}}`