sin theta = 8/17 and theta is in the second quadrant. Find sin(2theta),cos(2theta),tan(2theta) exactly
sin(2theta)
cos(2theta)
tan(2theta)
sin(2theta) would it be 2 x (8/17)
cos(2theta) would be 2 x (15/17)
tan(2theta) would be 2 x (8/17 divided by 15/17)
is this correct? I tried punching that into my homework for as an attempt and it comes back incorrect. Maybe I am not writing the exact answers correctly?
To find the exact values of sin(2theta), cos(2theta), and tan(2theta), we can use the double-angle formulas for trigonometric functions.
Given that sin(theta) = 8/17 and theta is in the second quadrant, we can use the Pythagorean identity to find cos(theta).
sin(theta) = 8/17
cos(theta) = sqrt(1 - sin^2(theta))
cos(theta) = sqrt(1 - (8/17)^2)
cos(theta) = sqrt(1 - 64/289)
cos(theta) = sqrt(225/289)
cos(theta) = 15/17
Now, we can use the double-angle formulas to find sin(2theta), cos(2theta), and tan(2theta).
sin(2theta) = 2sin(theta)cos(theta)
sin(2theta) = 2 * (8/17) * (15/17)
sin(2theta) = 240/289
cos(2theta) = cos^2(theta) - sin^2(theta)
cos(2theta) = (15/17)^2 - (8/17)^2
cos(2theta) = 225/289 - 64/289
cos(2theta) = 161/289
tan(2theta) = sin(2theta) / cos(2theta)
tan(2theta) = (240/289) / (161/289)
tan(2theta) = 240/161
tan(2theta) = 1.49
So, the exact values of sin(2theta), cos(2theta), and tan(2theta) are:
sin(2theta) = 240/289
cos(2theta) = 161/289
tan(2theta) = 1.49
To find the exact values of sin(2theta), cos(2theta), and tan(2theta), we need to use the double-angle formulas for trigonometric functions.
First, let's find sin(2theta):
sin(2theta) = 2 * sin(theta) * cos(theta)
We know that sin(theta) = 8/17, and since theta is in the second quadrant, which is negative for cosine, we can use the Pythagorean identity to find cos(theta):
cos(theta) = -√(1 - sin^2(theta))
= -√(1 - (8/17)^2)
= -√(1 - 64/289)
= -√(225/289)
= -15/17
Now substitute the values into the formula for sin(2theta):
sin(2theta) = 2 * (8/17) * (-15/17)
= -240/289
So, sin(2theta) is equal to -240/289.
Next, let's find cos(2theta):
cos(2theta) = cos^2(theta) - sin^2(theta)
cos(2theta) = (-15/17)^2 - (8/17)^2
= (225/289) - (64/289)
= 161/289
So, cos(2theta) is equal to 161/289.
Finally, let's find tan(2theta):
tan(2theta) = sin(2theta) / cos(2theta)
tan(2theta) = (-240/289) / (161/289)
= -240/161
= -240/161
= -240/161
So, tan(2theta) is equal to -240/161.
In summary, the exact values of sin(2theta), cos(2theta), and tan(2theta) for the given situation are:
- sin(2theta) = -240/289
- cos(2theta) = 161/289
- tan(2theta) = -240/161
Please note that double-checking the calculations is always a good idea, so it's important to verify these results.
sinθ = 8/17 in QII means that
cosθ = -15/17
sin2θ = 2sinθcosθ = 2(8/17)(-15/17)
cos2θ = cos^2θ - sin^2θ = (15/16)^2 - (8/17)^2
tan2θ is of course, sin2θ/cos2θ