Given that tan25 degrees =a,express tan205 in terms of a.
How do 25° and 205° relate?
Well, 205 - 180 = 25
and since 205° is in quadrant III, we know from the CAST rule
that the tangent in III is positive, therefore
tan 205°
= tan(205-180)°
= tan 25°
= a
To find tan(205 degrees) in terms of a, we can use the property that tan(x + 180 degrees) = tan(x).
First, we need to determine the equivalent angle to 205 degrees within the range of 0 to 180 degrees.
To do this, we subtract 180 degrees from 205 degrees:
205 degrees - 180 degrees = 25 degrees
Since we already know that tan(25 degrees) = a, we can express tan(205 degrees) in terms of a as follows:
tan(205 degrees) = tan(25 degrees + 180 degrees) = tan(25 degrees)
Therefore, tan(205 degrees) = a.
To express tan 205 degrees in terms of a, we can use the tangent addition formula:
tan (A + B) = (tan A + tan B) / (1 - tan A * tan B)
In this case, we have A = 180 degrees and B = 25 degrees. Therefore, tan 205 degrees can be expressed as:
tan 205 degrees = tan (180 degrees + 25 degrees)
First, let's find the value of tan 180 degrees and tan 25 degrees using the given information:
tan 25 degrees = a
Now, tan 180 degrees is undefined because the tangent function has vertical asymptotes at odd multiples of 90 degrees. But we know that tangent is an odd function and has periodicity of 180 degrees. Therefore, we can rewrite tan 180 degrees as:
tan 180 degrees = tan (180 degrees - 180 degrees)
Now, we can use the tangent addition formula to express tan 205 degrees in terms of a:
tan 205 degrees = (tan 180 degrees + tan 25 degrees) / (1 - tan 180 degrees * tan 25 degrees)
Substituting the values:
tan 205 degrees = (undefined + a) / (1 - undefined * a)
Since tan 180 degrees is undefined, the expression becomes:
tan 205 degrees = a / (1 - undefined * a)
But undefined times any value is still undefined, so the expression simplifies to:
tan 205 degrees = a / (1 - undefined)
Finally, since anything divided by 1 is itself:
tan 205 degrees = a
Therefore, tan 205 degrees can be expressed in terms of a as simply a.