Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {27}Exercise 12Find the exact value of tan15 ◦ .Putting A = B = θ in the expansion formula for tan(A + B), we obtaintan2θ =2tanθ1 − tan 2 θ .Exercise 13Show that t = tan67 1 2◦satisfies the quadratic equation t 2 − 2t − 1 = 0 and hence find itsexact value.The angle between two linesThe tangent expansion formula can be used to find the angle, or rather the tangent of theangle, between two lines.ynγlPβOαxSuppose two lines l and n with gradients m 1 and m 2 , respectively, meet at the point P.The gradient of a line is the tangent of the angle it makes with the positive x-axis. So, ifl and n make angles α and β, respectively, with the positive x-axis, then tanα = m 1 andtanβ = m 2 . We will assume for the moment that α > β, as in the diagram above.Now, if γ is the angle between the lines (as shown), then γ = α − β. Hencetanγ = tan(α − β) =tanα − tanβ1 + tanα tanβ = m 1 − m 21 + m 1 m 2,provided m 1 m 2 ≠ −1. If m 1 m 2 = −1, the two lines are perpendicular and tanγ is notdefined.In general, the above formula may give us a negative number, since it may be the tangentof the obtuse angle between the two lines.