Solving Trig Equations | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Solving Trig Equations

• You can use the symmetry of trigonometric graphs to find multiple solutions to a trigonometric equation.

How are trigonometric equations of the form sin x = k solved?

• The equation sinx = 0.5 yields solutions in the range 0° < x < 360°, which are x = 30° and x = 150°.
  • To confirm, one can check on a calculator that both sin(30°) and sin(150°) result in 0.5.
• The first solution is obtained by taking the inverse sine of 0.5, resulting in x = sin⁻¹(0.5) = 30°.
• The second solution stems from the graph's symmetry of y = sinx between 0° and 360°.
  • Sketch the graph and draw a vertical line from x = 30° to the curve, then horizontally across to another point on the curve, and finally vertically back to the x-axis.
  • Due to the curve's symmetry, the new value of x is 180° − 30° = 150°.
• Generally, if x° is an acute angle solving sinx = k, then 180° − x° is the obtuse angle solving the same equation.
• If the calculator presents x as a negative value, continue drawing the curve to the left of the x-axis to aid visualization.

How are trigonometric equations of the form cos x = k solved?

• In the range 0° < x < 360°, the solutions to the equation cosx = 0.5 are x = 60° and x = 300°.
  • To confirm, one can verify on a calculator that both cos(60°) and cos(300°) yield 0.5.
• The first solution is obtained by taking the inverse cosine of 0.5, resulting in x = cos⁻¹(0.5) = 60°.
  • The second solution arises from the graph's symmetry of y = cosx between 0° and 360°.
• Sketch the graph and draw a vertical line from x = 60° to the curve, then horizontally across to another point on the curve, and finally vertically back to the x-axis.
  • Due to the curve's symmetry, the new value of x is 360° − 60° = 300°.
• Generally, if x° is an angle solving cosx = k, then 360° − x° is another angle solving the same equation.
• If the calculator displays x as a negative value, continue drawing the curve to the left of the x-axis to facilitate understanding.

How are trigonometric equations of the form tan x = k solved?

• The equation tan x = 1 yields solutions of x = 45° and x = 225° within the range of 0° to 360°.
  • Confirming these solutions on a calculator, both tan(45°) and tan(225°) result in 1.
• The first solution, x = 45°, is directly obtained by taking the inverse tangent of both sides: x = tan⁻¹(1).
• The second solution, x = 225°, arises from the symmetry of the tangent graph within the range of 0° to 360°.
• To illustrate this symmetry, sketch the graph of y = tan x.
• Draw a vertical line from x = 45° to intersect the curve, then extend it horizontally to another point on the curve, representing a different "branch" of tan x.
  • Extend a vertical line from this point back to the x-axis.
  • This new x-value is found by adding 180° to the initial x-value: 45° + 180° = 225°.
• In general, for any angle x° that solves the equation tan x = k, adding 180° yields another solution.
• If the calculator provides a negative value for x, continuing to draw the curve to the left of the x-axis can assist in visualizing the symmetry.