Question

Find sin 2x, cos 2x, and tan 2x if tanx=-3 and x terminates in quadrant IV. sin 2x cos 2x tan 2x

Answer 1

To find sin 2x, cos 2x, and tan 2x given that tan x = -3 and x terminates in quadrant IV, apply the double-angle trigonometric identities and calculate the values using the assigned conditions for the tangent of x and the quadrant.

Step-by-step explanation:

To find sin 2x, cos 2x, and tan 2x given that tan x = -3 and x terminates in quadrant IV, we can use the trigonometric identities:

In quadrant IV, sin x is negative, and cos x is positive. Since tan x = sine x / cosine x, we can find sin x and cos x using the Pythagorean Theorem where tan²x + 1 = sec²x, and sec x is the reciprocal of cos x. Solving for sin x and cos x yields sin x = -3 / √(3²+1) and cos x = 1 / √(3²+1). We can then substitute these values into the identities above to find sin 2x, cos 2x, and tan 2x.

Answer 2

The trigonometric values for x in quadrant IV with tanx = -3 are calculated using trigonometric identities: sin 2x = -3/5, cos 2x = -4/5, and tan 2x = 3/4.

Step-by-step explanation:

To find sin 2x, cos 2x, and tan 2x given that tanx = -3 and x terminates in quadrant IV, we first use the trigonometric identity tan 2x = 2 tan x / (1 - tan2 x). Since tanx is given as -3, we can calculate tan 2x as 2*(-3) / (1 - (-3)2) = -6 / (1 - 9) = -6 / (-8) = 3/4.

To find sin 2x and cos 2x, we use the fact that sin2 x + cos2 x = 1 and the identities for double angles: sin 2x = 2 sin x cos x and cos 2x = cos2 x - sin2 x. However, we need the values of sin x and cos x, which we can find using the Pythagorean identity for tan x being the ratio of sin x over cos x. We are given that tan x equals -3, which means sin x / cos x = -3. As x is in quadrant IV, where sine is negative and cosine is positive, we assume sin x = -3k and cos x = k for some positive k. Using the Pythagorean identity, (-3k)2 + k2 = 1 results in 10k2 = 1, giving us k = 1/√10. This results in sin x = -3/√10 and cos x = 1/√10. Now we can calculate sin 2x = 2 * (-3/√10) * (1/√10) = -6/10 = -3/5 and cos 2x = (1/√10)2 - (-3/√10)2 = (1/10) - (9/10) = -8/10 = -4/5.