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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Answer 1

Starting from tan x, you can find sec x, because of the trigonometric identity 1 + tan² x = sec² x.

 

1 + (-4/3)² = 1 + 16/9 = 25/9 = sec² x.

 

So sec x = ±√(25/9) = ±5/3.  But since x is in Quadrant II, sec x has to be negative.  That's because sec x has the same sign as cos x, because sec x = 1 / cos x.  We know that cos x is negative is Quadrant II, therefore so is sec x.  So sec x = -5/3.

 

Since sec x and cos x are reciprocals of each other, cos x = 1/sec x = -3/5.

 

Now use the identity sin² x + cos² x = 1, to find sin x:

 

sin² x + (-3/5)² = 1

sin² x + 9/25 = 1

sin² x = 16/25

sin x = ±4/5

 

Again, we know that sin x is positive in Quadrant II, so sin x = 4/5.

 

Now that we know sin x and cos x, we can use the double angle formulas to find sin 2x and cos 2x.

 

sin 2x = 2 sin x cos x = 2 (4/5) (-3/5) = -24/25

cos 2x = cos² x - sin ² x = (-3/5)² - (4/5)² = 9/25 - 16/25 = -7/25

 

Finally use the identity tan x = sin x / cos x to find tan 2x:

 

tan 2x = sin 2x / cos 2x = (-24/25) / (-7/25) = -24/-7 = 24/7