find the value of tan37.5 degree
To find the value of tan(37.5 degrees), you can use the tan double angle formula, which states that tan(2θ) = (2 * tan(θ)) / (1 - tan²(θ)).
In this case, we can rewrite 37.5 degrees as the sum of two angles: 30 degrees and 7.5 degrees.
Using the double angle formula, we can find the value of tan(30 degrees) first:
tan(30 degrees) = (2 * tan(15 degrees)) / (1 - tan²(15 degrees)).
Then, we can find the value of tan(15 degrees):
tan(15 degrees) = sin(15 degrees) / cos(15 degrees).
From trigonometric identities, we know sin(15 degrees) = √[(1 - cos(30 degrees)) / 2]
and cos(15 degrees) = √[(1 + cos(30 degrees)) / 2].
By substituting these values into the equation for tan(15 degrees), we get:
tan(15 degrees) = √[(1 - cos(30 degrees)) / 2] / √[(1 + cos(30 degrees)) / 2].
Substituting the values of tan(15 degrees) back into the equation for tan(30 degrees), we obtain:
tan(30 degrees) = (2 * (√[(1 - cos(30 degrees)) / 2] / √[(1 + cos(30 degrees)) / 2])) / (1 - (√[(1 - cos(30 degrees)) / 2] / √[(1 + cos(30 degrees)) / 2]))².
This gives us the value of tan(30 degrees), which is needed to find the value of tan(37.5 degrees).
Finally, we can substitute the value of tan(30 degrees) into the double angle formula:
tan(37.5 degrees) = (2 * tan(30 degrees)) / (1 - tan²(30 degrees)).
37.5 = 75/2 = 60 + 30/2
so, tan 37.5 = tan 75/2
= (1-cos75)/sin75
= (1-cos(60+15))/sin(60+15)
... you get the idea