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Trigonometric Graphs and Equations

Pythagoras and Trigonometry

Why there is more than one answer

  • A question gives an interval, almost always 0° ⩽ x ⩽ 360°, and expects every solution inside it
  • The calculator's inverse function returns just one angle, so the others come from the symmetry of the graph
The same idea on the cosine curve: the calculator gives 60 degrees, and the curve reaches that same height again at 360 − 60 = 300 degrees, so the cosine pair is symmetrical about 360 rather than 180
Source: Solving trig equations by Save My Exams
  • Two answers is the usual number over a 360° interval for sine or cosine, and the answer line normally shows two spaces
Finding the second solution of a sine equation from the graph: the calculator returns 30 degrees where the curve first reaches 0.5, and reading horizontally across to where the curve reaches 0.5 again gives 180 − 30 = 150 degrees by symmetry
Source: Solving trig equations by Save My Exams
  • Tangent behaves differently: its solutions are 180° apart rather than mirrored
The tangent curve with the first solution at 45 degrees and the second found by adding 180 to give 225 degrees, because each branch of the curve repeats the one before it exactly 180 degrees along
Source: Solving trig equations by Save My Exams

The method

  • Rearrange first, so the equation reads sin x = …, cos x = … or tan x = …, treating sin x as if it were a single unknown
  • Take the inverse function to get the first angle
  • Sketch the relevant graph across the interval and draw the horizontal line at that value
  • Read off where else the line cuts the curve, using the symmetry rules
    • For sine, the second solution is 180° − the first
    • For cosine, the second solution is 360° − the first
    • For tangent, add 180° to move to the next branch
  • Where the first angle comes out negative or outside the interval, add or subtract 360° to bring it inside before using the symmetry
  • Check each answer by substituting it back
Worked example

A sine equation with a negative value

Solve 3 sin x + 2 = 0 for 0° ⩽ x ⩽ 360°. Give your answers correct to 1 decimal place.

Solution:

  • Rearrange to isolate sin x: 3 sin x = −2, so sin x = −2 ÷ 3
  • The calculator gives sin⁻¹(−0.6666…) = −41.8103…, which is outside the interval
  • Sine is negative below the axis, so both solutions lie between 180° and 360°
  • Using the symmetry about 270°, the solutions are 180° + 41.8103… and 360° − 41.8103…
  • So x = 221.8° or x = 318.2° to 1 decimal place
  • Check: sin 221.8° and sin 318.2° both give −0.667 to 3 decimal places
Worked example

A cosine equation

Solve 4 cos x = 1 for 0° ⩽ x ⩽ 360°. Give your answers correct to 1 decimal place.

Solution:

  • Divide by 4: cos x = 0.25
  • The calculator gives cos⁻¹(0.25) = 75.5224…
  • For cosine the second solution is 360° minus the first
  • 360 − 75.5224… = 284.4775…
  • So x = 75.5° or x = 284.5° to 1 decimal place
  • Check: the two answers are symmetrical about 180°, as the cosine curve requires

Reading a reflex answer

  • A question may give the value and ask specifically for the reflex angle, meaning the one between 180° and 360°
  • Work out the calculator answer first, then apply the symmetry rule to reach the reflex partner
  • This is the same method, simply asking for the second solution instead of both