If tan(A+B) = 2 and tan B =1/3, find tan A
tan(A+B) = (tanA + tanB)/(1-tanAtanB)
sub tanB=1/3 and solve for tanA
you should get tanA = 1
I'm not tan expert, but I can tell you this much - if you're looking for tan A, maybe it went on vacation without leaving a note. But seriously, let's solve this math problem together.
We know that tan(A + B) = 2 and tan B = 1/3. In order to find tan A, we need to do a bit of trigonometric manipulation.
Using the formula for tan (A + B):
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
Substituting the given values:
2 = (tan A + (1/3)) / (1 - (1/3) * tan A)
Now, let's simplify. Cross-multiplying and rearranging:
2(1 - (1/3) * tan A) = tan A + (1/3)
Now, time to distribute the 2:
2 - (2/3) * tan A = tan A + (1/3)
Let's isolate the tan A terms:
-(2/3) * tan A - tan A = 1/3 - 2
Combining like terms:
-(5/3) * tan A = -5/3
Dividing both sides by -(5/3):
tan A = 1
There you have it! tan A = 1. So it looks like tan A didn't go on vacation after all.
To find tan A, we can use the formula for tan(A+B):
tan(A+B) = (tan A + tan B) / (1 - tan A * tan B)
Given that tan(A+B) = 2 and tan B = 1/3, we can substitute these values into the formula:
2 = (tan A + 1/3) / (1 - (1/3) * tan A)
Next, let's simplify the equation:
2 - 2/3 * tan A = tan A + 1/3
To get rid of the fractions, we can multiply the entire equation by 3:
6 - 2 * tan A = 3 * tan A + 1
Now, let's rearrange the equation to isolate the term involving tan A:
6 - 1 = 3 * tan A + 2 * tan A
5 = 5 * tan A
Finally, divide both sides of the equation by 5:
tan A = 1
Therefore, tan A = 1.
To find the value of tan A, we can make use of the given formula for tan(A + B). Recall the formula:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
From the given information, we have tan(A + B) = 2 and tan B = 1/3. Plugging these values into the formula, we get:
2 = (tan A + 1/3) / (1 - (tan A * 1/3))
To solve this equation, we can start by multiplying both sides by the denominator (1 - (tan A * 1/3)):
2 * (1 - (tan A * 1/3)) = tan A + 1/3
Expanding the left side:
2 - 2 * tan A / 3 = tan A + 1/3
Next, we can move all terms containing tan A to one side of the equation:
2 - 1/3 = tan A + 2 * tan A / 3
5/3 = 5/3 * tan A
Now, we can isolate tan A by dividing both sides of the equation by 5/3:
tan A = (5/3) / (5/3) = 1
Therefore, tan A = 1.