Find the value of tan2x, (pi/2)<x<pi, given secx=-5/4.
So cosx=-4/5? I've no clue what to do next.
According to the given domain, the angle must be in quadrant II,
We are given cosx = - 4/5.
remember that cosine is adjacent/hypotenuse, so construct a right angles triangle with base of 4 and hypotenuse of 5, you should recognize the 3-4-5 rightangled triangle so the height (or opposite) is 3 and
sinx = 3/5
remember that tangent = sine/cosine
and tanx = (3/5)/(-4/5) = - 3/4
have you come across the identiy
tan 2x = 2tanx/(1 - tan^2 x) ?
tan2x = 2(-3/4)/(1 - 9/16)
= (-3/2)/(7/16)
= -24/7
Thanks!
To find the value of tan2x, we'll need to use trigonometric identities to express tan2x in terms of secx.
First, let's use the Pythagorean identity to find sinx:
sin^2x + cos^2x = 1
sin^2x + (-4/5)^2 = 1
sin^2x + 16/25 = 1
sin^2x = 1 - 16/25
sin^2x = 9/25
sinx = √(9/25)
sinx = 3/5 [since sinx is positive in the given range]
Now, we can use the identity tanx = sinx/cosx to find tanx:
tanx = (3/5) / (-4/5)
tanx = -3/4
Now, we'll use the double-angle identity for tangent:
tan2x = 2tanx / (1 - tan^2x)
tan2x = 2(-3/4) / (1 - (-3/4)^2)
tan2x = -6/4 / (1 - 9/16)
tan2x = -3/2 / (16/16 - 9/16)
tan2x = -3/2 / 7/16
tan2x = (-3/2) * (16/7)
tan2x = -24/7
Therefore, the value of tan2x, where (π/2) < x < π and secx = -5/4, is -24/7.
To find the value of tan(2x), we need to use the double-angle identity for tangent. The formula for tan(2x) is:
tan(2x) = (2tan(x))/(1-tan²(x))
Since we are given the value of sec(x), we can derive the value of tan(x) using the reciprocal identities. Recall that:
sec(x) = 1/cos(x)
So, if sec(x) = -5/4, we can substitute this value to find cos(x):
1/cos(x) = -5/4
To solve for cos(x), we can cross-multiply:
4 = -5cos(x)
Now, divide both sides by -5:
cos(x) = -4/5
Now that we have cos(x), we can use it to find tan(x). Recall that:
tan(x) = sin(x)/cos(x)
To find sin(x), we can use the Pythagorean identity. Since:
cos²(x) + sin²(x) = 1
We can substitute the value of cos(x) we just found:
(-4/5)² + sin²(x) = 1
16/25 + sin²(x) = 1
sin²(x) = 1 - 16/25
sin²(x) = 9/25
sin(x) = ±sqrt(9/25) = ±3/5
Now we can find the value of tan(x):
tan(x) = sin(x)/cos(x) = (±3/5)/(-4/5) = ±3/4
Finally, we can substitute the value of tan(x) into the formula for tan(2x) to calculate:
tan(2x) = (2tan(x))/(1-tan²(x))
= (2(±3/4))/(1 - (±3/4)²)
Since x is between π/2 and π, we know that tan(x) is negative. Therefore, we can substitute -3/4 into the formula:
tan(2x) = (2(-3/4))/(1 - (-3/4)²)
= (-3/2)/(1 - 9/16)
Next, simplify the expression:
tan(2x) = (-3/2)/(16/16 - 9/16)
= (-3/2)/(7/16)
= -24/14
Finally, simplify the fraction:
tan(2x) = -12/7