Find sin(2theta) if tan of theta = 7/24, pi < theta < 3pi/2 ?
If tanØ = 7/24 and Ø is in III
then sinØ = -7/25 and cosØ = -24/25
sin (2Ø) = 2sinØcosØ
= 2(-7/25)(-24/25) = 336/625
To find sin(2theta), we can use the double-angle formula for sine:
sin(2theta) = 2 * sin(theta) * cos(theta)
First, we need to find the value of sin(theta) and cos(theta) using the given information.
Given: tan(theta) = 7/24, pi < theta < 3pi/2
We know that tangent is equal to the ratio of sine to cosine:
tan(theta) = sin(theta) / cos(theta)
Rearranging the equation, we get:
sin(theta) = tan(theta) * cos(theta)
Since we already have the value of tan(theta), we can substitute it into the equation:
sin(theta) = (7/24) * cos(theta)
Now we need to find the value of cos(theta). We can use the Pythagorean identity for tangent:
tan^2(theta) + 1 = sec^2(theta)
Substituting the given value of tangent, we get:
(7/24)^2 + 1 = sec^2(theta)
49/576 + 1 = sec^2(theta)
49/576 + 576/576 = sec^2(theta)
625/576 = sec^2(theta)
Taking the square root of both sides, we get:
sqrt(625/576) = sec(theta)
25/24 = sec(theta)
Now we have both sin(theta) and cos(theta). We can substitute these values into the double-angle formula for sine:
sin(2theta) = 2 * sin(theta) * cos(theta)
= 2 * ((7/24) * cos(theta)) * cos(theta)
= (14/24) * cos^2(theta)
= (7/12) * cos^2(theta)
Therefore, sin(2theta) = (7/12) * cos^2(theta)
To find sin(2theta), we need to use the double-angle formulas for sine.
The double-angle formulas state that sin(2theta) = 2 * sin(theta) * cos(theta).
To find sin(theta), we can use the given information that tan(theta) = 7/24.
Since tan(theta) = opposite/adjacent, we can set up a right triangle using the values for opposite and adjacent sides.
Let x represent the opposite side of the triangle, and y represent the adjacent side.
Based on the given information, we have tan(theta) = x/y = 7/24.
Using the Pythagorean theorem, we know that x^2 + y^2 = hypotenuse^2.
Since we are given that pi < theta < 3pi/2, it means theta lies in the third quadrant of the unit circle.
In the third quadrant, sin(theta) is negative. So, we can write sin(theta) = -x/hypotenuse.
Using the Pythagorean theorem and the given information, we can solve for x and y:
x^2 + y^2 = hypotenuse^2,
(x/24)^2 + y^2 = (7/24)^2.
Simplifying the equation:
x^2 + 576y^2 = 49.
Since sin(theta) is -x/hypotenuse and we're given that theta lies in the third quadrant, sin(theta) = -x/hypotenuse = -x.
To find x, we can solve the equation:
x^2 + 576(24^2 - x^2) = 49,
x^2 + 576(576 - x^2) = 49,
x^2 + 576^2 - 576x^2 = 49,
-575x^2 + 576^2 = 49,
-575x^2 = 49 - 576^2,
x^2 = (49 - 576^2)/-575.
Simplifying the equation and taking the square root of both sides:
x = sqrt((49 - 576^2)/-575).
Now that we have the value of x, we can calculate sin(theta):
sin(theta) = -x = -sqrt((49 - 576^2)/-575).
Finally, we can calculate sin(2theta) using the double-angle formula:
sin(2theta) = 2 * sin(theta) * cos(theta).
Since we have sin(theta) and tan(theta), we can use the Pythagorean identity to find cos(theta):
cos^2(theta) + sin^2(theta) = 1,
cos^2(theta) = 1 - sin^2(theta),
cos^2(theta) = 1 - ((-x)^2).
Solving for cos(theta):
cos(theta) = sqrt(1 - ((-x)^2)).
Now we can substitute these values into the double-angle formula to find sin(2theta):
sin(2theta) = 2 * sin(theta) * cos(theta),
sin(2theta) = 2 * (-sqrt((49 - 576^2)/-575)) * sqrt(1 - ((-x)^2)).