4-8 The graphs of sine, cosine and tangent

y = sin x and y = cos x

• Both oscillate between −1 and +1 with period 360° (2π rad); the cosine curve is the sine curve shifted left by 90°.
• Sine passes through (0, 0), peaks at 90°, returns to zero at 180°, minimum at 270°, and completes at 360°.
• Key points to plot: 0°, 90°, 180°, 270°, 360° — five points are enough for one cycle.

y = tan x

• Period 180° (π rad), range all reals, with vertical asymptotes at x = 90° + 180°k where cosine vanishes; the curve crosses zero at 0°, 180°, 360°.
• Between two asymptotes the tangent increases continuously from −∞ to +∞.

Reading graphs

• Solutions of an equation such as sin x = ½ on 0° ≤ x < 360° are the x-coordinates where the curve cuts the line y = ½ — here x = 30° and x = 150°.
• Number of solutions equals the number of intersection points, so always state the interval.