what is tan2a if sina=1/2
if sinA = 1/2
from your basic angle ratios, you know that A = 30° or 150°
then 2A = 60° or 300°
tan 2A = tan 60 = √3
or
tan 2A = tan 300° = -√3
sina = (1/2)
cosa = √( 1 - (sina)^2)
= (√3)/2
tana = sina/cosa = 1/√3
tan2a = (2tana/(1-(tana)^2)
Plug in the value of tan(a).
Well, sine of a is 1/2, so a must be an angle in the first quadrant. Now, let's find tan(2a) using a bit of trigonometry humor.
Tan(2a) can be expressed as sin(2a) / cos(2a). But before we go there, have you ever heard about the periodic table of elements? Well, in the world of trigonometry, we have something similar called the unit circle. It's like a party circle where all the trigonometric functions are dancing! So, let's join the party and find the values of sin(2a) and cos(2a).
Since sine is 1/2, we know that the point (1/2, √3/2) is on the unit circle. And since cos is positive in the first quadrant, the x-coordinate 1/2 tells us that cos(2a) is also 1/2.
Now, let's move on to sin(2a). Remember the double angle formula? It states that sin(2a) = 2sin(a)cos(a). Plugging in the value of sin(a) as 1/2 and cos(a) as 1/2, we get sin(2a) = 2 * (1/2) * (1/2) = 1/2.
Now that we have both sin(2a) and cos(2a), we can find tan(2a) by dividing sin(2a) by cos(2a).
So, tan(2a) = (1/2) / (1/2) = 1.
Therefore, tan(2a) = 1 when sin(a) = 1/2.
And that's the funny answer for you! Keep laughing, my friend!
To find the value of tan(2a) given that sin(a) = 1/2, we can use the double-angle identity for the tangent function.
The double-angle identity for tangent is:
tan(2a) = (2tan(a)) / (1 - tan^2(a))
First, let's find the value of tan(a) using the given value of sin(a).
Given sin(a) = 1/2, we know that sin(a) = opposite/hypotenuse in a right triangle. This means that if we let the opposite side be 1 and the hypotenuse be 2, we can find the adjacent side using the Pythagorean theorem:
adjacent^2 = hypotenuse^2 - opposite^2
adjacent^2 = 2^2 - 1^2
adjacent^2 = 3
adjacent = sqrt(3)
Therefore, tan(a) = opposite/adjacent = 1/sqrt(3) = sqrt(3)/3.
Now, substitute this value into the double-angle identity:
tan(2a) = (2tan(a)) / (1 - tan^2(a))
tan(2a) = (2(sqrt(3)/3)) / (1 - (sqrt(3)/3)^2)
tan(2a) = (2sqrt(3)/3) / (1 - 3/9)
tan(2a) = (2sqrt(3)/3) / (6/9 - 3/9)
tan(2a) = (2sqrt(3)/3) / (3/9)
tan(2a) = (2sqrt(3)/3) * (9/3)
tan(2a) = 6sqrt(3)/9
tan(2a) = 2sqrt(3)/3
Therefore, tan(2a) is equal to 2sqrt(3)/3.
To find the value of tan(2a) given that sin(a) = 1/2, we can use the double-angle formula for tangent:
tan(2a) = (2 * tan(a)) / (1 - tan^2(a))
Let's first find the value of tan(a) using the given information that sin(a) = 1/2.
We know that sin(a) = opposite/hypotenuse in a right triangle. Since sin(a) = 1/2, we can assume that the opposite side of angle a is 1 and the hypotenuse is 2.
Using the Pythagorean theorem, we can find the adjacent side:
adjacent^2 + opposite^2 = hypotenuse^2
adjacent^2 + 1^2 = 2^2
adjacent^2 + 1 = 4
adjacent^2 = 3
adjacent = sqrt(3)
Now, we can find the value of tan(a) using the opposite and adjacent sides:
tan(a) = opposite/adjacent
tan(a) = 1/sqrt(3)
To simplify this, we rationalize the denominator by multiplying both the numerator and denominator by sqrt(3):
tan(a) = (1/sqrt(3)) * (sqrt(3)/sqrt(3))
tan(a) = sqrt(3)/3
Now that we have found the value of tan(a), we can substitute it into the double-angle formula for tangent:
tan(2a) = (2 * tan(a)) / (1 - tan^2(a))
tan(2a) = (2 * sqrt(3)/3) / (1 - (sqrt(3)/3)^2)
tan(2a) = (2 * sqrt(3)/3) / (1 - 3/9)
tan(2a) = (2 * sqrt(3)/3) / (6/9 - 3/9)
tan(2a) = (2 * sqrt(3)/3) / (3/9)
tan(2a) = (2 * sqrt(3)/3) * (9/3)
tan(2a) = (2 * sqrt(3) * 9) / (3 * 3)
tan(2a) = (2 * 3 * 3 * sqrt(3)) / (3 * 3)
tan(2a) = (18 * sqrt(3)) / 9
tan(2a) = 2 * sqrt(3)
Therefore, tan(2a) is equal to 2 * sqrt(3) or approximately 3.464.