Solving Basic Trig Equations
What are basic trig equations?
- The types of equations dealt with here are those of the form
- where is a constant
- (Most) scientific calculators will give you an answer to this type of equation
- using inverse sin/cos/tan
- e.g.
- but trig equations usually have more than one solution
- using inverse sin/cos/tan
- For inverse sin (sin-1) and inverse tan (tan-1)
- (most) calculators will give an answer between -90° and 90°
- For inverse cos (cos-1)
- (most) calculators will give an answer between 0° and 180°
- A question may require answers beyond these ranges
- Typically, answers between 0° and 360° are required
- The properties (in particular, symmetry) of the trig graphs allow us to find all answers to trig equations
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/when a calculator gives x as a negative value when using inverse sin
- extend your graph sketch for negative values of x (i.e. to the left of the y-axis)
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
- (Most) calculators will not give x as a negative value when using inverse cos
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/when a calculator gives x as a negative value when using inverse tan
- extend your graph sketch for negative values of x (i.e. to the left of the y-axis)
Exam Tip
- Use a calculator to check your solutions by substituting them into the original equation
- e.g. 60° is a correct solution of cos x = 0.5 as cos 60° = 0.5 on a calculator
but 330° is an incorrect solution as cos 330° ≠ 0.5
- e.g. 60° is a correct solution of cos x = 0.5 as cos 60° = 0.5 on a calculator
Worked example
Solve sin x = 0.25 in the range 0° < x < 360°, giving your answers correct to 1 decimal place
Use a calculator to find the first solution (by taking inverse sin of both sides)
x = sin-1(0.25) = 14.4775… = 14.48° to 2 dp
Sketch the graph of y = sin x and mark on (roughly) where x = 14.48 and y = 0.25 would be
Draw a vertical line up to the curve, then horizontally across to the next point on the curve, then vertically back down to the x-axis
Find this value using the symmetry of the curve (by taking 14.48 away from 180)
180° – 14.48° = 165.52°
Give both answers correct to 1 decimal place
x = 14.5° or x = 165.5°