find exact value using sum or differnce equations. tan(19pi/12)
Sometimes students find it easier to think in degrees
19π/12 radians = 285°
skipping the degree symbol ...
and 285 = 240+45
we know tan 240 = +tan60 = √3 by the CAST rule
tan 45 = 1
tan(285)
= tan(240 + 45)
= (tan240 + tan45)/(1 - tan240tan45)
= (√3+1)/(1-√3)
tan(19π/12) = (√3+1)/(1-√3)
Why did the triangle go to the party alone?
Because it couldn't find its tangent or cosecant!
Alright, let's find the exact value of tan(19π/12) using sum and difference identities.
We know that tan(π/4) = 1, so let's rewrite 19π/12 as the sum of π/4 and 9π/6:
19π/12 = (3π/4) + (9π/6)
Now, we can use the tangent sum formula:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
Let's plug in the values:
tan(19π/12) = (tan(3π/4) + tan(9π/6)) / (1 - tan(3π/4)tan(9π/6))
Now, we know that tan(3π/4) = 1 and tan(9π/6) = tan(3π/2) = undefined.
So, the denominator becomes 1 - (1)(undefined), which is also undefined.
Therefore, the exact value of tan(19π/12) using sum and difference equations is undefined.
Sometimes math can be a little tricky, but don't worry, there's always a solution around the corner!
To find the exact value of tan(19π/12), we can use the sum and difference formulas for tangent.
The sum formula for tangent is:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
Using this formula, we can rewrite tan(19π/12) as:
tan( π/4 + π/3)
We know that tan(π/4) = 1 and tan(π/3) = √3.
Now, substituting these values into the sum formula, we have:
tan( π/4 + π/3) = (tan(π/4) + tan(π/3)) / (1 - tan(π/4)tan(π/3))
= (1 + √3) / (1 - 1 * √3)
Simplifying further by rationalizing the denominator, we get:
= (1 + √3) / (1 - √3)
To rationalize, multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + √3):
= [(1 + √3)*(1 + √3)] / [(1 - √3)*(1 + √3)]
= (1 + 2√3 + 3) / (1 - 3)
= (4 + 2√3) / (-2)
= -(2 + √3)
Therefore, the exact value of tan(19π/12) is -(2 + √3).
To find the exact value of tan(19π/12) using sum or difference equations, we need to express the angle in terms of angles for which we know the tangent values.
We start by converting 19π/12 to degrees:
(19π/12)(180/π) = 285 degrees
Since 285 degrees is not a special angle for which we know the tangent value, we'll use a difference equation to rewrite it.
The most commonly used difference equations for finding exact values of tangent involve 45-degree angles.
To express 285 degrees as a difference between 45-degree angles:
285 degrees = (6 * 45 degrees) + 15 degrees
Now, we can use the difference identities for tangent:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
Applying the equation to our case:
tan(285 degrees) = tan[(6 * 45 degrees) + 15 degrees]
Using the difference identity for tangent:
tan(285 degrees) = (tan(6 * 45 degrees) + tan(15 degrees)) / (1 - tan(6 * 45 degrees) * tan(15 degrees))
Now, we need to find the tangent values for the angles involved.
tan(45 degrees) = 1 (a known value)
To find tan(15 degrees), we can use the half-angle identity for tangent:
tan(A/2) = (1 - cos A) / sin A
tan(30 degrees) = (1 - cos(60 degrees)) / sin(60 degrees)
For sin(60 degrees), we know it is √3 / 2
For cos(60 degrees), we know it is 1/2
tan(30 degrees) = (1 - 1/2) / (√3 / 2) = (√3 - 1) / √3
Now, we have:
tan(285 degrees) = (tan(6 * 45 degrees) + tan(15 degrees)) / (1 - tan(6 * 45 degrees) * tan(15 degrees))
= (tan(270 degrees) + (√3 - 1) / √3) / (1 - tan(270 degrees) * (√3 - 1) / √3)
Now, we can substitute the known values:
tan(285 degrees) = (−∞ + (√3 - 1)/√3) / (1 - −∞ * (√3 - 1)/√3)
= (−∞ + (√3 - 1)/√3) / (1 + (√3 - 1)/√3)
The simplified form will be the exact value of tan(19π/12), expressed in terms of the square root of 3 and 1.