If Cosinex = 4/5 and x has its terminal side in quadrant lV, find the exact value of tangent 2x
Ifcosx = 4/5, then sinx = -3/5 in IV by Pythagoras.
tan 2x = sin 2x/cos 2x
= 2sinxcosx/(cos^2x - sin^2x)
= 2(-3/5)(4/5)/(16/25 - 9/25)
= (-24/25)/(7/25)
= (-24/25)(25/7) = -24/7
To find the exact value of tangent 2x, we need to use the double-angle identity for tangent:
tangent 2x = 2 * tangent x / (1 - tangent^2 x)
Given that Cosinex = 4/5, we can use the Pythagorean identity to find the value of sine x:
sine x = √(1 - cosine^2 x)
= √(1 - (4/5)^2)
= √(1 - 16/25)
= √(9/25)
= 3/5
Since x has its terminal side in quadrant IV, where cosine is positive and sine is negative, we have:
cosine x = 4/5
sine x = -3/5
Now, we can substitute the values of sine x and cosine x into the double-angle identity to find the value of tangent 2x:
tangent 2x = 2 * (tangent x) / (1 - (tangent x)^2)
First, let's find the value of tangent x:
tangent x = sine x / cosine x
= (-3/5) / (4/5)
= -3/4
Now, substitute this value into the double-angle identity:
tangent 2x = 2 * (-3/4) / (1 - (-3/4)^2)
= -6/4 / (1 - 9/16)
= -6/4 / (7/16)
= -6/4 * (16/7)
= -96/28
= -24/7
Therefore, the exact value of tangent 2x is -24/7.