tanx= -12/5 x in Q 2.

triangle is 5, 12, 13

To find the values of sin(2x), cos(2x), and tan(2x), we can use the double-angle identities. These identities relate the trigonometric ratios of twice an angle to the trigonometric ratios of the angle itself.

First, we need to find the value of x in the second quadrant (Q2) using the given equation:

tan(x) = -12/5

Since tangent is negative in the second quadrant, we know that tan(x) = -12/5. We can use the inverse tangent function (arctan) to find the angle x:

x = arctan(-12/5)

Next, we can substitute the value of x into the double-angle identities to find sin(2x), cos(2x), and tan(2x):

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

Now, let's calculate these values step by step.

Step 1: Finding x in Q2
Using a calculator or math software, calculate the inverse tangent of -12/5:

x ≈ arctan(-12/5) ≈ -67.38 degrees (rounded to two decimal places)

Step 2: Finding sin(2x)
Using the value of x we found in step 1, we can find sin(x) and cos(x):

sin(x) = sin(-67.38°) ≈ -0.9238795325 (rounded to 10 decimal places)
cos(x) = cos(-67.38°) ≈ 0.3826834324 (rounded to 10 decimal places)

Now, substitute these values into the double-angle identity for sin(2x):

sin(2x) = 2sin(x)cos(x) ≈ 2(-0.9238795325)(0.3826834324) ≈ -0.7071067812 (rounded to 10 decimal places)

So, sin(2x) ≈ -0.7071067812 (rounded to 10 decimal places).

Step 3: Finding cos(2x)
Using the values of sin(x) and cos(x) we found in step 2, we can calculate cos(2x) using the double-angle identity:

cos(2x) = cos^2(x) - sin^2(x) ≈ (0.3826834324)^2 - (-0.9238795325)^2 ≈ 0.9238701195 (rounded to 10 decimal places)

So, cos(2x) ≈ 0.9238701195 (rounded to 10 decimal places).

Step 4: Finding tan(2x)
Using the value of tan(x) and the values of sin(x) and cos(x) we found earlier, we can calculate tan(2x) using the double-angle identity:

tan(2x) = (2tan(x))/(1 - tan^2(x)) ≈ (2(-12/5))/(1 - (-12/5)^2) ≈ -0.2298850575 (rounded to 10 decimal places)

So, tan(2x) ≈ -0.2298850575 (rounded to 10 decimal places).

To summarize:
sin(2x) ≈ -0.7071067812 (rounded to 10 decimal places)
cos(2x) ≈ 0.9238701195 (rounded to 10 decimal places)
tan(2x) ≈ -0.2298850575 (rounded to 10 decimal places)