SOLUTION: Please help me with this equation. Find sin(2x), cos(2x), and tan(2x) from the given information. tan(x) = −4/3, x in Quadrant II sin(2x) = cos(2x) = tan(2x) =

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Question 1206406: Please help me with this equation.
Find sin(2x), cos(2x), and tan(2x) from the given information.
tan(x) = −4/3, x in Quadrant II
sin(2x) =

cos(2x) =

tan(2x) =

Found 3 solutions by math_tutor2020, Theo, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

I'll replace x with the Greek letter theta (symbol ) since it is more commonly used with trig angles.
Also it's to avoid potential upcoming confusion when I'll use x in a different way.

Given: where angle is in quadrant II.

Goal: Compute , , and

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Quadrant II is the northwest quadrant i.e. upper left corner.
This is for any point (x,y) such that x < 0 and y > 0.

Here's what the diagram looks like

Tangent is the ratio of opposite over adjacent.

This is the same as saying
y = opposite
x = adjacent

If then we would have y = 4 and x = -3 so the x < 0 and y > 0 requirements are fulfilled.
The terminal point is located at (x,y) = (-3,4).

Now let's find the hypotenuse of this right triangle using the Pythagorean theorem.

The hypotenuse is 5 units long.
We have a 3-4-5 right triangle.

y = opposite = 4
x = adjacent = -3
r = hypotenuse = 5

which means,

and

Furthermore,

which is one of the answers
and also
see note below

which is another answer
lastly
see note below

which is the last answer we need

Note: There are other identities you could use as an alternative pathway.
Refer to this list of trig identities
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf
Specifically refer to the "double angle formulas" section on page 2.

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A summary of the answers:

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
you are given that tan(x) = -4/3 and x is in the second quadrant.

tan is opposite divided by adjacent.
that makes opposite = 4 and adjacent = -3
hypotenuse = sqrt(4^2 + (-3)^2) = 5.

in the graph of angles, x represents the side adjacent to the angle and y represents the side opposite the angle.

in the second quadrant, x is minus and y is positive.
tan(x) = -4/3 is opposite / adjacent is y/x.
y is positive and x is negative, so -(4/3) is 4/-3, and not -3/4.
since -4/3 is equivalent to 4/-3 is equivalent to -(4/3), you can use either one if the formula allows it.

the simplest way to find your answer is to use the double angle identity formulas.

tan(2x) = 2tan(x) / (1-tan^2(x))
since tan(x) = -4/3 in the second quadrant, you get:
tan(2x) = 2 * -4/3) / (1 - (-4/3)^2) = 3.428571429.
the answer in simplified fraction form would be 24/7.

sin(2x) = 2 * sin(x) * cos(x)
since sin(x) = 4/5 in the second quadrant and cos(x) = -3/5 in the second quadrant, you get:
sin(2x) = 2 * 4/5 * -3/5 = -.96
the answer in simplified fraction form would be -24/25.

cos(2x) = cos^2(x) - sin^2(x)
since sin(x) = 4/5 in the second quadrant and cos(x) = -3/5 in the second quadrant, you get:
cos(2x) = (-3/5)^2 - (4/5)^2 = -.28
the answer in simplified fraction form would be -7/25.

you can confirm these answers are good in the following manner.
you are given that tan(x) = -4/3 in the second quadrant.
use your calculator to solve for the angle to get angle = -53.13010235 degrees.
convert that to a positive equivalent angle by adding 360 to it to get 306.8690976 degrees.
that's in the fourth quadrant.
the equivalent angle in the first quadrant is 360 minus that = 53.13010235 degrees.
the equivalent angle in the second quadrant is 180 minus that = 126.8698976 degrees.

to confirm this is the correct angle, find tan(126.8698976) = -1.33333.....
convert to fraction to get -4/3, which is correct because that's what was given.

you have x = 126.8698976 degrees.
2x is therefore equal to 253.7397953 degrees.

sin (that) = -.96
cos(that) = -.28
tan(that) = 3.428571429.

these values agree with what we got earlier using the double angle formulas, so the answer is confirmed to be good.

the double angle identity formulas can be found in the following reference.

https://en.wikipedia.org/wiki/List_of_trigonometric_identities under the heading of Multiple-angle and half-angle formulae

Answer by MathTherapy(10552)   (Show Source):
You can put this solution on YOUR website!

Please help me with this equation. Find sin(2x), cos(2x), and tan(2x) from the given information. tan(x) = −4/3, x in Quadrant II sin(2x) = cos(2x) = tan(2x) = , with x in Quadrant II As this is a PYTHAGOREAN TRIPLE (3-4-5), r = 5. Therefore,