what is the exact value of tan 105 degrees
-2-√3
105 = 60+45
tan 105° = (√3+1)/(1-√3)
To find the exact value of tan 105 degrees, we can make use of the identity:
tan(180° - θ) = -tan(θ)
In this case, we can write:
tan 105° = -tan(180° - 105°)
which simplifies to:
tan 105° = -tan 75°
We can further simplify this:
tan 75° = tan(30° + 45°)
Using the identity:
tan(α + β) = (tan α + tan β) / (1 - tan α * tan β)
We can substitute α = 30° and β = 45°:
tan 75° = (tan 30° + tan 45°) / (1 - tan 30° * tan 45°)
The exact values of tan 30° and tan 45° are √3/3 and 1, respectively. Substituting these values:
tan 75° = (√3/3 + 1) / (1 - (√3/3)(1))
Simplifying further, we get:
tan 75° = (√3 + 3) / (3 - √3)
To find the exact value of tan 105 degrees, we can use the trigonometric identity:
tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
Here, we can rewrite 105 degrees as the sum of 90 degrees and 15 degrees:
105 degrees = 90 degrees + 15 degrees
Now, let's find the values of tan 90 degrees and tan 15 degrees.
The tangent of 90 degrees is undefined because at that angle, the value of the cosine function is zero. Therefore, we cannot find the exact value of tan 105 degrees using this identity.
However, we can use another trigonometric identity to find the value of tan 15 degrees:
tan 15 degrees = sin 15 degrees / cos 15 degrees.
Now, we need to consult a table of trigonometric values or use a calculator to find the exact values of sin 15 degrees and cos 15 degrees.
Using these values, we can then divide sin 15 degrees by cos 15 degrees to find the exact value of tan 15 degrees.
If you have a calculator, you can directly input "tan 105" and it will give you an approximate value of tan 105 degrees.