Question

Given sinθ = -(7/25) and pi < θ < 3 times (pi/2), what is the exact solution of sin 2θ?49/625336/625527/625576/625

Answer 1

Explanation

We are given the following:

`\sin\theta=-(7)/(25)\text{ }and\text{ }\pi<\theta<(3\pi)/(2)`

We are required to determine the exact value of sin 2Θ.

We know that the trigonometric identity for sin 2Θ is thus:

`\sin2\theta=2\sin\theta\cos\theta`

Since Θ is between 180° and 270° as given above, we know that this angle falls in the third quadrant, and sine and cosine are negative in this quadrant.

Therefore, we have:

`\begin{gathered} \sin\theta=(7)/(25)\to(opposite)/(hypotenuse) \\ \\ \text{ Using the Pythagorean theorem,} \\ hypotenuse^2=opposite^2+adjacent^2 \\ 25^2=7^2+adj^2 \\ adj^2=25^2-7^2 \\ adj=√(25^2-7^2) \\ adj=√(625-49)=√(576) \\ adj=24 \\ \\ \text{ Hence, we have:} \\ \cos\theta=(adjacent)/(hypotenuse) \\ \cos\theta=(24)/(25) \\ \\ \text{ In the third quadrant, } \\ \cos\theta=-(24)/(25) \end{gathered}`

Now, we can determine the value of sin 2Θ as:

`\begin{gathered} \sin2\theta=2\sin\theta\cos\theta \\ \sin2\theta=2\cdot(-(7)/(25))\cdot(-(24)/(25)) \\ \sin2\theta=(2*7*24)/(25*25) \\ \sin2\theta=(336)/(625) \end{gathered}`

Hence, the answer is:

`\sin2\theta=(336)/(625)`