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5.1 SOLVING TRIGONOMETRIC EQUATIONS
USING GRAPHING TECHNOLOGY
PM12
All trigonometric equations can be solved using graphing technology.
A. PREDICTING SOLUTIONS

Given the equation sinx = 0.45, solve to 2 decimal places over the domain 0  x  2
STEPS:
1) Sketch a sine wave on the grid for the domain 0  x  2 .
2) Draw the line y = 0.45 on the same grid.
3) a) Within the domain, how many solutions do you predict? ______
b) What quadrants are the solution(s) in? ______
B. SOLVING USING GRAPHING CALCULATOR

Given the equation sinx = 0.45, solve to 2 decimal places over the domain 0  x  2
STEPS:
Set the window
to the domain
[xmin, xmax]
given
1) Enter Y1 = sinx
2) Enter Y2 = 0.45
3) Sketch the graph & find the intersection. Each root is the intersection of the
two functions.
4) Write the roots of the equation within the domain ( 0  x  2 ): ________
NOTE: The GENERAL SOLUTION is an expression that represents all roots of the equation.
Ex. Give the general solution for sinx = 0.45
_____________
Principles Math 12 Section 5.1 Page 3
EXAMPLE 1 : Given the equation: cos2x = -0.35
a) Draw a rough sketch over the domain 0  x  2 of y = cos2x and y = -0.35
b) How many solutions are there for the equation cos2x = -0.35 over the domain
0  x  2 ? ____________
c) Solve the equation cos2x = -0.35 for x, where 0  x  2 __________________
d) Give the general solution for the equation cos2x = -0.35
_______________________
Often asked as
“solve over the
real numbers”
NOTE: What does the period have to do with the general solution?
EXAMPLE 2 : For each equation, determine the number of solutions within the domain
0  x  2 .
Determine the period of the function first and do a rough sketch
a) cosx = 0
b) sin2x = -0.5
c) cos3x =0.5
d) tan2x = 3
EXAMPLE 3 : Determine the number of solutions for the equation sinbx = sin bx 
where 0  x  2 .
Principles Math 12 Section 5.1 Page 4
1
2
EXAMPLE 4 : Solve the equation 1 + sin2x = -1 – 3sin2x over the domain 0  x  2 .
a) What is the period of each function: i) 1 + sin2x _____ and ii) -1 – 3sin2x _____
b) How many solutions do we expect to find in given domain 0  x  2 ? _______
c) Solve the equation graphically below. State appropriate window dimensions.
Y1 =
Y2=
[
xmin
,
xmax
]
[
ymin
,
ymax
]
X = _____________________________________
d) Give the general solution for the equation 1 + sin2x = -1 – 3sin2x where the domain is the
set of all real numbers.
X = ______________________________________
SOLVING & WRITING THE GENERAL SOLUTION:
STEPS:
1) State each function you are graphing
2) Roughly sketch the graph over one period (If you don’t know the
period then sketch the graph over 0 ≤ x ≤ 2π to help you find it)
3) Write the general solution: root + n(period)
Principles Math 12 Section 5.1 Page 5
EXAMPLE 5 : Solve and write the general solution for each equation:
a) cos(x + π) =
1
2


b) cos (x  1)  cos x
2
2
c) 2 sin 2 x  sin x  2  0
EXAMPLE 6 : Solve the equation 2sinx = x
EXAMPLE 7 : Solve the equation tanx + sinx = 2 where 0 ≤ x ≤ 2π.
ASSIGNMENT: p. 302 # (1, 3, 5-8, 10)odds, 11, 12, (13, 14, 19)*
Principles Math 12 Section 5.1 Page 6