Tan 165°
Use appropriate identity and evaluate
165
= 180-15 or
= 120+45 or
= 90+75
I will pick 120+45
tan 165 = tan(120+45)
= (tan120 + tan 45)/(1 - tan120tan45) **
2
we know tan 45=1
tan120 = -tan60 , by the CAST rule
= -√3 , from the 30-60-90 triangle
so back in **
tan165 = (-√3 + 1)/(1 - (-√3)(1))
= (1 - √3)/(1 + √3)
Thank you
To find the value of tan 165°, we can use the following identity:
tan(180° - θ) = -tan(θ)
Since 165° = 180° - 15°, we can rewrite tan 165° as:
tan 165° = -tan 15°
Now we need to find the value of tan 15°. We can use the following identity:
tan(α + β) = (tan α + tan β) / (1 - tan α * tan β)
By using tan(45° + 30°):
tan 15° = (tan 45° + tan 30°) / (1 - tan 45° * tan 30°)
We know that tan 45° = 1 and tan 30° = 1/√3, so substituting these values in:
tan 15° = (1 + 1/√3) / (1 - 1 * 1/√3)
Simplifying:
tan 15° = (√3 + 1) / (√3 - 1)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator (√3 + 1):
tan 15° = (√3 + 1) * (√3 + 1) / (√3 - 1) * (√3 + 1)
Expanding and simplifying:
tan 15° = (3 + 2√3 + 1) / (3 - 1)
tan 15° = (4 + 2√3) / 2
tan 15° = 2 + √3
Therefore, tan 165° = -tan 15° = - (2 + √3).
To find the value of tan 165°, we can use the tangent addition formula.
The tangent addition formula states that tan(A + B) = (tan A + tan B) / (1 - tan A * tan B).
In this case, we can rewrite 165° as the sum of two angles, 90° and 75°.
So, tan 165° = tan (90° + 75°).
Now, let's find the tangent values for 90° and 75°.
We know that tan 90° = undefined because it is the value at which the tangent function has vertical asymptotes.
Next, we need to find the tangent of 75°.
To do this, we can use the tangent addition formula again:
tan (90° + 75°) = (tan 90° + tan 75°) / (1 - tan 90° * tan 75°).
Substituting the values we have, we get:
tan (90° + 75°) = (undefined + tan 75°) / (1 - undefined * tan 75°).
Now, find the value of tan 75°. Use a calculator or reference table to determine that tan 75° is approximately 2.747477.
Plugging this value back into the formula, we get:
tan (90° + 75°) = (undefined + 2.747477) / (1 - undefined * 2.747477).
Since tan 90° is undefined, our final answer is undefined.
Therefore, tan 165° is undefined.