tan(theta)=(7/24), sin(theta)<0
find cos(2theta)
If tan is positive and sin is negative, theta is in the third quadrant. From Pythagorean considerations, sin theta = -7/25 and cos theta = -24/25. (Imagine a 7,25,25 right triangle)
Use the double angle formula for
cos(2 theta)
= cos^2(theta) - sin^2(theta)
= (24^2-7^2)/25^2 = 527/625
2theta will be in the first quadrant, 32.5 degrees
Check: theta = 180 + 16.26 = 196.26 deg
2 theta = 392.50 => 32.5 degrees
Change the third sentence to:
(Imagine a 7,24,25 right triangle)
To find the value of cos(2θ), we can use the double-angle formula for cosine:
cos(2θ) = 2cos^2(θ) - 1
To use this formula, we first need to find the value of cos(θ).
Given that tan(θ) = 7/24, we can use the definition of tangent to find the values of sin(θ) and cos(θ).
tan(θ) = sin(θ)/cos(θ)
7/24 = sin(θ)/cos(θ)
To find sin(θ) and cos(θ), we can use a Pythagorean identity involving sin(θ), cos(θ), and the unit circle.
sin^2(θ) + cos^2(θ) = 1
Since we are given that sin(θ) < 0, we can substitute -sin(θ) for sin(θ) in the equation.
(-sin(θ))^2 + cos^2(θ) = 1
sin^2(θ) + cos^2(θ) = 1
Now we have two equations:
7/24 = (-sin(θ))/cos(θ)
sin^2(θ) + cos^2(θ) = 1
Using the first equation, we can solve for sin(θ) in terms of cos(θ):
7/24 = (-sin(θ))/cos(θ)
Multiply both sides by cos(θ):
(7/24)cos(θ) = -sin(θ)
Divide both sides by cos(θ):
sin(θ) = -(7/24)cos(θ)
Now we can substitute this expression for sin(θ) in the second equation:
(-[(7/24)cos(θ)])^2 + cos^2(θ) = 1
Simplifying this equation:
(49/576)cos^2(θ) + cos^2(θ) = 1
Multiply both sides by 576 to eliminate the fraction:
49cos^2(θ) + 576cos^2(θ) = 576
Combine like terms:
625cos^2(θ) = 576
Divide both sides by 625:
cos^2(θ) = 576/625
Taking the square root of both sides, and noting that cos(θ) is positive in the given quadrant (since sin(θ) < 0), we get:
cos(θ) = √(576/625) = 24/25
Now that we have found the value of cos(θ), we can substitute it into the double-angle formula:
cos(2θ) = 2cos^2(θ) - 1
Substituting cos(θ) = 24/25:
cos(2θ) = 2(24/25)^2 - 1
Evaluating this expression:
cos(2θ) = 2(576/625) - 1
cos(2θ) = 1152/625 - 1
cos(2θ) = 1152/625 - 625/625
cos(2θ) = (1152 - 625)/625
cos(2θ) = 527/625
Therefore, cos(2θ) = 527/625.