SOLUTION: Find the exact values of sin(2θ), cos(2θ), and tan(2θ) when sin θ =-7/12 and 180∘ < 𝜃 < 270∘.

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Question 1162249: Find the exact values of sin(2θ), cos(2θ), and tan(2θ) when sin θ =-7/12 and 180∘ < 𝜃 < 270∘.
Found 3 solutions by Theo, Edwin McCravy, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
here's how i think it's going to work.
you have sin(x) = -7/12
that makes x = -35.68533471.
add 360 to that to get the equivalent positive angle of 324.3146653.
the reference angle for that is 360 - that = 35.68533471.
the equivalent angle for that in the third quadrant is that plus 180 = 215.6853347.
take the sine of that and you get -7/12, which is the correct sine for angle x in the third quadrant.
you should be able to simply find sin(2x), cos(2x), tan(2x), by simply doubling that angle.
2 times 215.6853347 = 431.3706694.
you have:
x = 215.6853347
2x = 431.3706694
sin(2x) = .9476050057
cos(2x) = .3194444444.....
tan(2x) = 2.9664567.

if my assumptions are correct, those are your solutions.

using trigonometric identities, you shouls get the same answers.

you are given that sin(x) = -7/12
that makes x = 215.6853347, as we found out earlier.

the trigonometric identifies are:

sin(2x) = 2 * sin(x) * cos(x)
cos(2x) = cos^2(x) - sin^2(x)
tan(2x) = 2 * tan(x) / (1 - tan^2(x)

since we know x, we should be able to find those using the trig identities.

sin(x) = -.5833333333 stored in variable N
cos(x) = -,8133248621 stored in variable O
tan(x) = .7181848465 stored in variable P

i stored the values in those variables so i don't have to rewrite those values each time i use them.

the trigonometric identifies are rewritten using those variable names as shown below:

sin(2x) = 2 * N * O = .9476050057
cos(2x) = O^2 - N^2 = .3194444444
tan(2x) = 2 * P / (1 - P^2) = 2.96641567.

i get the same answers, so both methods seem to work.







Answer by Edwin McCravy(20063)   (Show Source): You can put this solution on YOUR website!
Theo apparently does not know the meaning of "exact value".  It means
NO DECIMALS!

[I corrected my typo for tan(θ), which was neither asked for nor needed.  My tan(2θ) was and is correct.]

 

We draw θ in QIII, with y (opposite side) =-7 and r (hypotenuse) = 12.
We calculate x (adjacent side):



Since x goes left of the origin, we take the negative answer, 




Then to calculate the trig ratios for 2θ, we will need






Edwin
Answer by MathTherapy(10556)   (Show Source): You can put this solution on YOUR website!
Find the exact values of sin(2θ), cos(2θ), and tan(2θ) when sin θ =-7/12 and 180∘ < 𝜃 < 270∘.
The other person who responded gave you WRONG answers. The problem asks for EXACT VALUES. What he gave you are NOT exact values!

. We then need to find A, or x, which will be negative, since  is an ∠ in the 3rd quadrant (180∘ < 𝜃 < 270∘).



Therefore, THREE of the 6 TRIG. RATIOS are: 

You may not need tan 𝜃, depending on which formula you use. 

For me, since I would've determined sin 2𝜃 and cos 2𝜃, I would then use the formula: . Then again, you decide!!



Below, I've done  for you. Follow that same concept to determine cos 2𝜃 and maybe tan 2𝜃.



 ------ Substituting , for, sin 𝜃, and  for cos 𝜃. 



BTW, Edwin, your tan 𝜃 is WRONG!! 

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