Download Solve sin(x) + 2 = 3 for 0° sin(x) + 2 = 3 sin(x) = 1 x = 90

SOLVING TRIGONOMETRIC EQAUTIONS
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Solve sin(x)
+ 2 = 3 for 0° < x < 360°
Isolate the variable-containing term:
sin(x) + 2 = 3
sin(x) = 1
x = 90°
•
Solve tan
2(x) + 3 = 0 for 0° < x < 360°
tan2(x) = –3
no solution
•
Solve
on 0° < x < 360°
To solve this, I need to factor.
x = 30°, 90°, 270°, 330°
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2(x) – sin(x) = 2 on 0° < x < 360°
Solve sin
This is a quadratic in sine, so I can factor:
sin2(x) – sin(x) – 2 = 0
(sin(x) – 2)(sin(x) + 1) = 0
sin(x) = 2 (not possible!) or sin(x) = –1
Only one of the factor solutions is sensible. For sin(x)
= –1, I get:
x = 270°
•
Solve cos
2(x) + cos(x) = sin2(x) on 0° < x < 360°
We have two variables so I need use a trig identity to get a quadratic in cosine:
2
2
cos (x) + cos(x) = sin (x)
2
2
cos (x) + cos(x) = 1 – cos (x)
2cos2(x) + cos(x) – 1 = 0
(2cos(x) – 1)(cos(x) + 1) = 0
cos(x) = 1/2 or cos(x) = –1
The first trig equation, cos(x) = 1/2, gives me x =
me x = 180°. So my complete solution is:
60° and x = 300°. The second equation gives
x = 60°, 180°, 300°
•
Solve sin(x)
= sin(2x) on 0° < x < 360°
sin(x) = 2sin(x)cos(x)
sin(x) – 2sin(x)cos(x) = 0
Factor sin( x)
sin(x)(1 – 2cos(x)) = 0
sin(x) = 0 or
1-2cos(x)=0
cos(x) = 1/2
x = 0°, 60°, 180°, 300°, 360°
•
Solve sin(x)
+ cos(x) = 1 on 0° < x < 360°
Square both sides.
2
2
(sin(x) + cos(x)) = (1)
2
2
sin (x) + 2sin(x)cos(x) + cos (x) = 1
2
2
[sin (x) + cos (x)] + 2sin(x)cos(x) = 1
1 + 2sin(x)cos(x) = 1
2sin(x)cos(x) = 0
sin(x)cos(x) = 0
x = 0°, 90°, 180°, 270°
Plugging back in.
sin(0°) + cos(0°) = 0 + 1 = 1
(this solution works)
sin(90°) + cos(90°) = 1 + 0 = 1
(this one works, too)
sin(180°) + cos(180°) = 0 + (–1) = –1
sin(270°) + cos(270°) = (–1) + 0 = –1
(oh;okay, so this one does NOT work)
(this one doesn't work, either)
So the actual solution is:
x = 0°, 90°
EXIT CARD.
WHAT DID I LEARN TODAY?
ASSESSMENT FOR YOUR LEARNING
____
a.
b.
1.
____
a.
b.
2.
____
a.
b.
3.
Which value for is a solution to ?
c.
d.
What quadrants do the solutions to lie in?
I, IV
c.
I, III
d.
III, IV
I, II
How many solutions does the equation have for ?
3
c.
4
d.
5
6
____ 4.
places.
a.
b.
Determine the related acute angle to the solutions of accurate to two decimal
____
Use the following graph of to estimate the solution of for .
a.
b.
5.
c.
d.
1.57
3.14
c.
d.
No solution
4.71
____
6.
Use the following graph of to estimate the solution(s) of for .
a.
b.
1.57, 4.71
3.14
c.
d.
0, 6.28
-1.57
____
a.
b.
7.
Factor the expression .
____
a.
b.
8.
Which of the following is NOT a solution to the equation for ?
c.
d.
____
a.
b.
9.
Which of the following is a solution for the equation ?
c.
d.
____
a.
b.
10.
Which is a solution for the equation ?
c.
d.
____
a.
b.
11.
Solve where .
,,
,,
____
a.
b.
12.
c.
d.
c.
d.
Which is a solution to the equation ?
c.
d.
ASSESSMENT FOR YOUR LEARNING
Answer Section
MULTIPLE CHOICE
no solution
,,
,,
1.
ANS: A
2.
ANS: C
3.
ANS: B
4.
ANS: A
5.
6.
ANS: D
ANS: A
7.
ANS: C
8.
9.
ANS: D
ANS: A
10.
ANS: A
11.
ANS: A
12.
ANS: B