Tan 75/2 solve this
tan[(90-15)/2]
tan [ 45 - 15/2 ]
but tan(a-b) = [tana-tanb]/[1+tana tan b]
so
[1-tan7.5]/[1+tan7.5]
what is tan 7.5 = tan 15/2 = tan 30/4
tan 15/2 = (1-cos 15)/sin 15
but
cos 15 = cos30/2 = sqrt[(1+cos30) /2]
we know cos 30 = sqrt(3)/2
so cos 15 = sqrt[(1+sqrt3/2)/2]
and
sin 15 = sin(30/2) = sqrt[(1-cos30)/2
so sin 15 = sqrt[(1-sqrt3/2)/2
so
tan 7.5 = {1-sqrt[(1+sqrt3/2)/2]}/ {sqrt[(1-sqrt3/2)/2}
now do
(1 -tan 7.5) / (1+tan 7.5)
tan 75 = tan(30+45)
= (tan30+tan45)/(1-tan30*tan45)
= (1/√3 + 1)/(1-1/√3*1)
= (√3+1)/(√3-1)
= (3+2√3+1)/2
= 2+√3
So, now you can figure
sin(75/2) and cos(75/2) and thus tan(75/2)
To solve the expression tan(75/2), we first need to understand that the angle given is in degrees.
The tangent function (tan) is a trigonometric function that relates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. In this case, we have an angle measure of 75 degrees divided by 2.
To find the tangent value, we can follow these steps:
1. Convert the angle from degrees to radians: Since most calculators work in radians, we need to convert the angle to radians. There are 2π radians in 360 degrees, so we can convert degrees to radians using the following formula:
Angle in radians = (Angle in degrees * π) / 180
Therefore, the angle in radians would be:
(75/2 * π) / 180
2. Use a scientific calculator: Now that we have the angle in radians, we can use a scientific calculator to evaluate the tangent function. Input the converted angle value into the calculator and find the tangent value.
The exact solution will depend on whether you want the result in decimal form or as a fraction, as well as the precision required.
Using the steps mentioned above, you can find the value of tan(75/2) by converting the angle to radians and then evaluating the tangent function on a calculator.