If it is given that tan(A-B)=tanA-tanB/1+tanAtanB and tanP-1/1+tanP=tan195 find p
To find the value of P, we will use the given equation:
tan(P - 1) / (1 + tan(P)) = tan(195)
Step 1: Simplify tan(195)
Since 195 is not a common angle, we need to rewrite 195 as a sum or difference of angles that we can work with.
We know that 180 degrees is equivalent to π radians, so we can use this conversion to find an angle we are more familiar with: 195 = 180 + 15.
So we have tan(195) = tan(180 + 15)
Using the trigonometric identity tan(A + B) = (tanA + tanB) / (1 - tanA*tanB), we can rewrite this as:
tan(180 + 15) = (tan(180) + tan(15)) / (1 - tan(180)*tan(15))
Step 2: Calculate tan(180) and tan(15)
Using known values, we can substitute in the tangents of 180 and 15.
tan(180) = 0
tan(15) = 0.267949
So our equation becomes:
tan(195) = (0 + 0.267949) / (1 - 0*0.267949)
Step 3: Simplify the expression
Since 0 multiplied by anything equals 0, we can further simplify the equation:
tan(195) = 0.267949 / 1
tan(195) = 0.267949
Step 4: Substitute into the original equation
Now we can substitute this result into the equation we are given and solve for P:
tan(P - 1) / (1 + tan(P)) = 0.267949
Simplifying the equation gives us:
(tan(P) - 1) / (1 + tan(P)) = 0.267949
Cross-multiplying gives us:
tan(P) - 1 = 0.267949 * (1 + tan(P))
Expanding the right side and simplifying further:
tan(P) - 1 = 0.267949 + 0.267949 * tan(P)
Combining like terms:
tan(P) - tan(P) * 0.267949 = 0.267949 + 1
Simplifying further:
tan(P) * (1 - 0.267949) = 1.267949
tan(P) * 0.732051 = 1.267949
Dividing both sides by 0.732051 gives us:
tan(P) = 1.731975
Step 5: Solve for P
To find P, we need to calculate the inverse tangent (arctan) of 1.731975.
P = arctan(1.731975)
Using a calculator or mathematical software, we find that
P ≈ 60 degrees
So, the value of P is approximately 60 degrees.
To find the value of P, we need to simplify the given equation: tan(P - 1) / (1 + tan(P)) = tan(195).
Step 1: Recall the identity for tan(A - B): tan(A - B) = (tanA - tanB) / (1 + tanA * tanB).
Comparing this identity with the given equation, we can see that P plays the role of A and 1 plays the role of B in the identity. Thus, we can write the equation as follows:
tan(P - 1) / (1 + tan(P)) = tan(P) - tan(1) / (1 + tan(P) * tan(1)) = tan(195).
Step 2: Simplify the equation further by substituting the values from the given equation.
tan(P) - tan(1) / (1 + tan(P) * tan(1)) = tan(195).
tan(1) can be evaluated as tan(1 degree) ≈ 0.017455 (using a calculator).
Hence, the equation becomes:
tan(P) - 0.017455 / (1 + tan(P) * 0.017455) = tan(195).
Step 3: Use the inverse tangent function (arctan or tan^(-1)) to solve for P.
arctan(tan(P) - 0.017455 / (1 + tan(P) * 0.017455)) = arctan(tan(195)).
P - 0.017455 / (1 + P * 0.017455) = 195.
Solving this equation will give you the value of P. You can use numerical methods such as Newton's method or trial and error to approximate the value.