1. 5.4 Equations and Graphs of Trigonometric Functions
Math 30-1

2. Determine the angle given the ratio.
Determine the solution(s) for the trigonometric equation
We seek the angle (the value of x) for which the cosine gives the ratio
Graphically
Reference Angle
The reference angle for
Quadrant I x = 30°
Quadrant IV x = 150°
Two solutions:
30° and 150°.
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3. Multiple Solutions
When the domain of the function is restricted (0° ≤ x < 360°), the solutions to the trigonometric equation must occur within that given restriction.
When the domain of the function is not restricted, often there are multiple solutions.
The solutions repeat themselves in multiples of 360° from each original solution.
The general solutions to the equation are x = 30° + 360°n or x = 150° + 360°n, n ϵ I
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4. Determine the solutions for the trigonometric equation
for the interval
Method 1: Solve Graphically
Graph the related function
The solutions are the x-intercepts of the graph of the related function.
The solutions for the interval 0° ≤ x < 360° are
x = 30°, 150°, 210°, 330°.
(0, 150)
(0, 210)
(0, 30)
(0, 330)
Math 30-1

5. Determine the solutions for the trigonometric equation
for the interval
30°
210°
Method 2: Solve algebraically
Quadrant I x = 30°
Quadrant III x = 150°
330°
150°
Quadrant I x = 30°
Quadrant III x = 150°
Math 30-1

6. Solving Trigonometric Equations
2sin2 x + 3sin x + 1 = 0
(2sin x + 1)(sin x + 1) = 0
2sin x + 1 = 0
sin x + 1 = 0
Reference
Angle
Math 30-1

7. Interpreting Graphs to Find Solutions
The diagram below shows the graphs of two trig functions
y = 4sin2x and y = 6sinx + 2, defined for 0 ≤ x ≤ 2p.
Describe how you could use this graph to estimate the
solution to the equation (4sin2x) (6sinx + 2) = 0 for 0 ≤ x ≤ 2p.
y = 6sinx + 2
y= 4sin2x
The solutions will be the points where the graphs intersect
the x-axis.
Therefore, the solutions are:
x = 0, 3.14, 6.28, 3.48, and 5.94

8. Interpreting Graphs to Find Solutions
Describe how you could use this graph to estimate the
solution to the equation 4sin2x = 6sinx + 2 for 0 ≤ x ≤ 2p.
y = 6sinx + 2
y= 4sin2x
The solutions will be the points where the graphs intersect.
Therefore, the solutions are:
x = and

9. Interpreting Graphs to Find Solutions
Use a graph to solve the equation
Therefore, the solutions are:
x = 0 and
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10. Using Technology to Find Solutions
Determine the lowest possible value of x, to the
nearest tenth for which
The smallest x-value is -2.9.
Math 30-1

11. Assignment Page 275 1, 3, 4a, c, 5c,d, 6a,c, 8, 12, 13, 14, 15, 19
Math 30-1