Solve the equation sin2x = tanx for 0° ≤ x ≤ 360°
sin2x = tanx Take everything to left hand side, so LHS equals 0sin2x - tanx = 0Recall double angle formula : sin2x = 2sinxcosx, also notice that tanx can be written as sinx/cosx giving us:2sinxcosx - sinx/cosx = 0 (gives us everything in terms of sinx and cosx)Multiply everything by cosx to get rid of the fraction2sinxcos2x -sinx = 0Notice sinx is common in both terms so we can now simplifysinx*(2cos2x-1) = 0This now means that either sinx =0 or 2cos2x-1 =0For sinx = 0sinx = 0, use inverse sin function on calculator and use CAST diagram or alternatively use y = sinx graph (in range 0° to 360° inclusive) to see where the graph cuts the x axis when y =0, which gives:x = 0°, 180°, 360°For 2cos2x-1 = 0cos2x= 0.5cosx = ±√0.5, which meanscosx = √0.5 and cosx =-√0.5Using inverse cos function on calculator of 0.5:x = 45°Plot x = 45° in cast diagram for positive cos and negative cos, which gives:x = 45°,135°,225°,315°Final answer: x = 0°, 180°, 360° or x = 45°,135°,225°,315°
