Use the sum or difference identity to find the exact value of tan 105 degree.
105=60+45
Use the sum identity.
To use the sum or difference identity to find the exact value of tan 105 degrees, we need to express 105 degrees as the sum/difference of two angles for which we know the exact values of the trigonometric functions.
Since tan is positive in the second and fourth quadrants, we need to find two angles in those quadrants whose tangents we know.
Let's express 105 degrees as the sum or difference of 90 degrees and another angle:
105 degrees = 90 degrees + 15 degrees
Now let's find the tangent of each angle:
tan 90 degrees is undefined because it corresponds to a vertical line in the unit circle.
However, we can find the tangent of 15 degrees:
tan 15 degrees = sin 15 degrees / cos 15 degrees
To find the exact value of sin 15 degrees, we can use a half-angle formula:
sin 15 degrees = sqrt[(1 - cos 30 degrees) / 2]
= sqrt[(1 - sqrt(3)/2) / 2]
= sqrt[(2 - sqrt(3))/4]
Similarly, we can find the exact value of cos 15 degrees using the half-angle formula:
cos 15 degrees = sqrt[(1 + cos 30 degrees) / 2]
= sqrt[(1 + sqrt(3)/2) / 2]
= sqrt[(2 + sqrt(3))/4]
Now, plug these values into the original tangent equation:
tan 15 degrees = sin 15 degrees / cos 15 degrees
= sqrt[(2 - sqrt(3))/4] / sqrt[(2 + sqrt(3))/4]
= sqrt[(2 - sqrt(3))/(2 + sqrt(3))]
So, the exact value of tan 105 degrees is sqrt[(2 - sqrt(3))/(2 + sqrt(3)).