5.5 Solving Trigonometric EquationsExample 1
A) Is a solution to ?
B) Is a solution to cos x = sin 2x ?
4
3x
1cos
2x
3x
Trigonometric Equations with a Single Trig Function
• For equations with a single trig function, isolate the trig function on one side.
• Solve for the variable by identifying the appropriate angles.
• Be prepared to express your answer in radian measure.
Other Strategies for Solving
Solving Trig Equations
• Put the equation in terms of one trig function (if possible).
• Solve for the trig function (using algebra – addition, subtraction, multiplication, division, factoring).
• Solve for the variable (using inverse trig functions, reference angles).
• Use a fundamental identity to end up with a single trig function.
Example 4
To solve an equation containing a single trig function:
Solve: 3sinx – 2 = 5sinx - 1* Isolate the function on one side of the equation. * Solve for the variable.
Solution: 3sinx - 5sinx = -1 +2 -2sinx = 1 sinx = -1/2 (Remember: x are the angles whose sine is -1/2) 7 11
: 2 and 26 6
Ans x n n
Example 5
Solve the equation on the interval [0 , 2π)
2 cos x − 1 = 0
2 cos x = 1 cos x =
x =
1
2
5,
3 3
Example 6 -Trigonometric Equations Quadratic in Form.
2Solve the equation: 2sin 3sin 1 0; 0 2x x x
Ans. π/6, π/2, 5π/6
Try to solve by factoringIt factors in the same manner as= (2x -1)(x – 1)
Solution: (2sinx – 1)(sinx -1) = 0 2sinx – 1 = 0 2sinx = 1 sinx = ½Therefore x = π/6, 5π/6
sinx – 1 = 0 sinx = 1 x = π/2
22 3 1x x
Example 8: Solve an Equation with a Multiple Angle.
Solve the equation: tan2 3 0 2x x
2 7 5: , , ,
6 3 6 3Ans
2Solve the equation: tan sin 3 tan ; 0 2x x x x
Ans. 0, π
Move all terms to one side, then factor out a common trig function.
Example 10
2Solve the equation: 2sin 3cos 0 0 2x x x
Ans. π/3, 5π/3
The equation contains more than one trig function; there is no common trig function. Try using an identity.
Example 11
Example 14 - using a calculator to solve
Solve the equation correct to four decimal places, 0 ≤ x ≤ 2π a. tan x = 3.1044 b. sin x = -0.2315
Ans. a. 1.2592, 4.4008 b. 3.3752, 6.0496
Use a calculator to find the reference angle, then use your knowledge of signs of trigonometric functions to find x in the required interval.