Lesson 5.3
Solving
Trigonometric Equations
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Solving Trigonometric Equations: A Comprehensive Guide with Examples, Slides of Algebra
Step 1: Isosolate cos x using algebraic skills. Step 2: Determine in which quadrants cosine is positive. Use the inverse function to assist by finding the angle ...
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Lesson 5.
Solving
Trigonometric Equations
Solving
Trigonometric Equations
To solve trigonometric equations:^ Use standard algebraic techniques learned in Algebra II.^ Look for factoring and collecting like terms.^ Isolate the trig function in the equation.^ Use the inverse trig functions to assist in determining^ solutions.
Solving
Trigonometric Equations^ 2 cos
1
0 x^ −
=
2 cos
1 x^ =^1 cos
2 x^ =
Solve:
x
π^
Step 1: Isosolate cos
x^ using algebraic skills.
Step 2: Determine in which quadrants cosine is positive. Use the inverse function to assist by finding the angle in Quad I first. Then use that angle as the reference angle for the other quadrant(s).
QI^
QIV
Note: cosine is positive in Quad I and Quad IV. Note: The reference angle is
π /3.
Solving
Trigonometric Equations
tan
x^ −
2
tan
x^ = 2
tan
x^
=^
Solve:
tan
x^ =
,^
,^
x
π^
π^
π^
Step 1:^ Step 2:
Note: Since there is a
±^ , all four quadrants
hold a solution with
π /4 being the reference
angle.
Q^
QII^
QIII
QIV
Solving
Trigonometric Equations^ tan
x^ +
2
sec
x^ −
3
3 tan
tan
x^
x
Try these:
1.^ 2.^ 3.
Solution
0,^
,^
,^ ,
x
π^
π^
π^
(^7) , 4
4 x
π^
π
,^
,^
x
π^
π^
π^
Solving
Trigonometric Equations
2 2sin
sin
1
0
x^
x −^
−^
=
(^
)(^
2sin
1 sin
x^
x +^
−^
2sin
1
0
sin^
1
0
x^
or^
x
+^
=^
−^
=
Solve:
1 sin^
2 x^ =^
−^
sin^
1 x^ =
7
(^11) , 6
6
x
π^
=^
2 x
π =
Factor the quadratic equation. Set each factor equal to zero.^ Solve for sin
x
Determine the correct quadrants for the solution(s).
Solving
Trigonometric Equations^ cos
sin
x^
x
+^
(^
)^
(^
) 2
2
cos^
1
sin x^
x +^
= 2
2
cos^
2 cos
1
sin
x^
x^
x
+^
+^ =
Solve:
2
2
cos^
2 cos
1
1
cos
x^
x^
x
+^
+^ =
−
2 2 cos
2 cos
0
x^
x +^
=
(^
2 cos
cos
1
0 x^
x^ +^
=
2 cos
cos^
x^
or^
x
=^
+^ =
cos^
x^ =^
cos^
1 x^ =^
−
x
π^
π
=^
x
Square both sides of the equation in order to change sine into terms of cosine giving only one trig function to work with. FOIL or Double Distribute Replace sin
2 x^ with 1 – cos
2 x
Set equation equal to zero since it is a quadratic equation. Factor Set each factor equal to zero.^ Solve for cos
x
Determine the solution(s).
X
Why is 3
π /2 removed as a solution?
It is removed because it does not check in the original equation.