How do you find (in radians) the exact value of:
tan[pi-(pi/12)]
tan(pi - pi/12),
Factor out pi:
tan(pi(1 - 1/12)),
tan(pi(11/12),
tan(11pi/12),
tan(11pi/12)*180/pi),
tan(165)deg.
Now, I'm not sure what else you want
done. The trig ratios(functions) cannot be written in radians or degrees, because they have no units.
The angles ONLY can be in degrees and radians.
To find the exact value of tan[π - (π/12)] in radians, we need to use some trigonometric identities and properties.
1. Start by finding the value of π/12 in radians:
π/12 is equal to (π/180) * 12, since there are 180 degrees in π radians.
Simplifying this, we get π/12 = π/15 radians.
2. Now, let's substitute the value of π/12 in the expression:
tan[π - (π/12)] becomes tan[π - (π/15)].
3. Use the difference identity for tangent:
The tangent difference identity states that tan(α - β) = (tan α - tan β) / (1 + tan α * tan β).
By substituting α = π and β = (π/15), we get:
tan[π - (π/15)] = (tan π - tan (π/15)) / (1 + tan π * tan (π/15)).
4. Applying the tangent values of π and π/15:
tan π is equal to 0, since the tangent of π radians is 0.
tan (π/15) is not a commonly known value, so we need to find a way to calculate it.
5. Use the tangent addition identity:
The tangent addition identity states that tan(α + β) = (tan α + tan β) / (1 - tan α * tan β).
By considering α = π/3 and β = (π/15), we get:
tan (π/3 + π/15) = (tan (π/3) + tan (π/15)) / (1 - tan (π/3) * tan (π/15)).
6. Simplify the equation:
The value of tan (π/3) is √3 (which can be derived using special triangles) and tan (π/15) is unknown.
So, the equation becomes: tan (4π/15) = (√3 + tan (π/15)) / (1 - √3 * tan (π/15)).
7. Solve for tan (π/15):
Rearrange the equation: (√3 - 1) * tan (π/15) = √3
Divide both sides by (√3 - 1): tan (π/15) = √3 / (√3 - 1).
8. Now that we have the value of tan (π/15), we can substitute it back into the original expression:
tan[π - (π/15)] = (0 - √3 / (√3 - 1)) / (1 + 0 * √3 / (√3 - 1))
= -√3 / (√3 - 1).
Therefore, the exact value of tan[π - (π/12)] in radians is -√3 / (√3 - 1).