# 0 therefore to solve tan θ a you must know which of

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0. Therefore, to solve tanθ=a, you must knowwhich of the two possible quadrants the angle is in and then make a correction if the angle isin the second or third quadrants. Fortunately, the correction is easy to remember: “just addπTo solve tanθ=a, these are the four possibilities.(1) tanθ=a,a >0 andθis in the first quadrant=θ= arctana(2) tanθ=a,a <0 andθis in the second quadrant=θ= arctana+π(3) tanθ=a,a >0 andθis in the third quadrant=θ= arctana+π(4) tanθ=a,a <0 andθis in the fourth quadrant=(θ= arctanaorθ= (arctana) + 2π
(2)sin(A±B) = sinAcosB±cosAsinB(3)cos(A±B) = cosAcosBsinAsinBMany of the other trig identities can be derived easily from these. For example, dividing (1)by cos2xgivestan2x+ 1 = sec2xAngleIn general, the angleθbetween the positivex-axis and the line segment from the origin to thepoint (a, b) – always measured in the counterclockwise direction – satisfiestanθ=ba.However, one can not simply concludeθ= arctanbawithout also checking quadrants.Example: Find the angle between thex-axis and line segment from (0,0) to (5,2).The answer isθ= tan-1(2

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