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Trigonometric Graphs and Equations

Pythagoras and Trigonometry

Why the graphs matter

  • Beyond a right-angled triangle the angle is no longer restricted to acute values, so sin, cos and tan are defined for every angle from 0° to 360° and beyond
  • The syllabus asks you to recognise, sketch and interpret y = sin x, y = cos x and y = tan x over 0° ⩽ x ⩽ 360°
  • The graphs are what make an equation solvable, because a calculator returns only one answer and the graph reveals the rest

Sine and cosine

  • Both are waves rising to 1 and falling to −1, and both repeat every 360°, which is their period
  • Plotting at every multiple of 90° is enough to get the shape right
x0°90°180°270°360°
sin x010−10
cos x10−101
  • y = sin x passes through the origin, while y = cos x starts at its peak, crossing the vertical axis at 1
The graph of y = sin x drawn from −360 to 360 degrees: a wave starting at the origin, rising to 1 at 90 degrees, back through zero at 180, down to −1 at 270 and back to zero at 360, repeating the same shape to the left of the axis
Source: Exact trig values by Save My Exams
  • The cosine curve is the sine curve shifted 90° to the left, so the two have the same shape in different positions
  • Sine is symmetrical about x = 90° and cosine about x = 180°, and those symmetries are what generate second solutions
The graph of y = cos x drawn from −360 to 360 degrees: the same wave shape as sine but starting at its peak of 1 on the vertical axis, crossing zero at 90 and 270 degrees and reaching −1 at 180
Source: Exact trig values by Save My Exams

Tangent

  • y = tan x breaks the pattern entirely: rather than one continuous wave it comes in separate branches, repeating every 180°, half the period of the other two
  • Each branch climbs from far below the axis to far above it
  • Vertical asymptotes sit at x = 90° and x = 270°, where the curve rises without ever reaching the line
  • The curve passes through the origin, and through (180°, 0) and (360°, 0)
  • Tangent has no value at 90° or 270°, which is why those are drawn as dotted lines rather than points
The graph of y = tan x drawn from −360 to 360 degrees: separate branches each climbing steeply from far below the axis to far above it, passing through zero at 0, 180 and 360 degrees, with dashed vertical asymptotes at 90 and 270 degrees and their negatives
Source: Exact trig values by Save My Exams
Worked example

Reading values off the graphs

Use the graphs to write down the value of sin 180°, cos 180° and the two values of x between 0° and 360° for which sin x = 0.5.

Solution:

  • The sine curve starts at 0, rises to 1 at 90°, returns to 0 at 180°, falls to −1 at 270° and returns to 0 at 360°
  • So sin 180° = 0
  • The cosine curve starts at 1, falls to 0 at 90°, reaches −1 at 180°, returns to 0 at 270° and back to 1 at 360°
  • So cos 180° = −1
  • For sin x = 0.5, draw the horizontal line y = 0.5 across the sine curve
  • It cuts the curve twice between 0° and 360°, both times where the curve is above the axis
  • The calculator gives the first: x = 30°
  • The sine curve is symmetrical about x = 90°, so the second is 180 − 30 = 150°
  • The two solutions are x = 30° and x = 150°