Solving Trig Equations (CIE IGCSE Maths: Extended)

You can use the symmetry of trig graphs to find multiple solutions to a trig equation

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

• The solutions to the equation sin x = 0.5 in the range 0° < x < 360° are x = 30° and x = 150°
• If you like, check on a calculator that both sin(30) and sin(150) give 0.5
• The first solution comes from your calculator (by taking inverse sin of both sides)
• x = sin^-1(0.5) = 30°
• The second solution comes from the symmetry of the graph y = sin x between 0° and 360°
• Sketch the graph
• Draw a vertical line from x = 30° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again
• By the symmetry of the curve, the new value of x is 180° - 30° = 150°
• In general, if x° is an acute angle that solves sin x = k, then 180° - x° is the obtuse angle that solves the same equation
• If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help

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

• The solutions to the equation cos x = 0.5 in the range 0° < x < 360° are x = 60° and x = 300°
• If you like, check on a calculator that both cos(60) and cos(300) give 0.5
• The first solution comes from your calculator (by taking inverse cos of both sides)
• x = cos^-1(0.5) = 60°
• The second solution comes from the symmetry of the graph y = cos x between 0° and 360°
• Sketch the graph
• Draw a vertical line from x = 60° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again
• By the symmetry of the curve, the new value of x is 360° - 60° = 300°
• In general, if x° is an angle that solves cos x = k, then 360° - x° is another angle that solves the same equation
• If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help

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

• The solutions to the equation tan x = 1 in the range 0° < x < 360° are x = 45° and x = 225°
• Check on a calculator that both tan(45) and tan(225) give 1
• The first solution comes from your calculator (by taking inverse tan of both sides)
• x = tan^-1(1) = 45°
• The second solution comes from the symmetry of the graph y = tan x between 0° and 360°
• Sketch the graph
• Draw a vertical line from x = 45° to the curve, then horizontally across to another point on the curve (a different “branch” of tan x), then vertically back to the x-axis again
• The new value of x is 45° + 180° = 225° as the next “branch” of tan x is shifted 180° to the right
• In general, if x° is an angle that solves tan x = k, then x° + 180° is another angle that solve the same equation
• If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help