Trig Equations with a Twist
Solve each equation on the interval 0 ≤ 𝑥 < 2𝜋 with exact values.
_________________ 1.
2 sin2 𝑥 − 5 sin 𝑥 + 2 = 0
_________________ 2.
sin2 𝑥 − 2 sin 𝑥 − 3 = 0
_________________ 3.
4 sin2 𝑥 − 3 = 0
Use identities to assist you in solving each of the following on the interval 0 ≤ 𝑥 < 2𝜋 with exact values.
_________________ 4.
2 sin 𝑥 = csc 𝑥
_________________ 5.
1 − 3 cos 𝑥 = sin2 𝑥
_________________ 6.
tan2 𝑥 = 2 sec 𝑥 − 1
_________________ 7.
cot 2 𝑥 = 3(csc 𝑥 − 1)
_________________ 8.
2 cos 2 𝑥 + sin 𝑥 = 1
_________________ 9.
2 cos 2 𝑥 − √3 cos 𝑥 = 0
_________________ 10. cos 2𝑥 + 3 cos 𝑥 − 1 = 0
_________________ 11. sin 2𝑥 = − sin 𝑥
_________________ 12. tan 2𝑥 = cot 𝑥
Consider the period and/or frequency of the graphs when solving each of the following on the interval
0 ≤ 𝑥 < 2𝜋 with exact values.
1
_________________ 13. sin 2𝑥 = 2
_________________ 14. cos 3𝑥 =
√2
2
_________________ 15. sin 3𝑥 = −
√3
2
Trig Equations with a Twist KEY
Solve each equation on the interval 0 ≤ 𝑥 < 2𝜋 with exact values.
𝜋 5𝜋
𝑥= ,
6 6
_________________
1. 2 sin2 𝑥 − 5 sin 𝑥 + 2 = 0
(2 sin 𝑥 − 1)(sin 𝑥 − 2) = 0
2 sin 𝑥 − 1 = 0 or sin 𝑥 − 2 = 0
3𝜋
𝑥=
2
_________________ 2.
1
sin 𝑥 = 2 or sin 𝑥 = 2
sin2 𝑥 − 2 sin 𝑥 − 3 = 0
(sin 𝑥 + 1)(sin 𝑥 − 3) = 0
sin 𝑥 + 1 = 0 or sin 𝑥 − 3 = 0
𝜋 2𝜋 4𝜋 5𝜋
𝑥= , ,
,
3 3 3 3
_________________
3.
sin 𝑥 = −1 or sin 𝑥 = 3
4 sin2 𝑥 − 3 = 0
4 sin2 𝑥 = 3
3
sin2 𝑥 = 4
sin 𝑥 = ±
√3
2
Use identities to assist you in solving each of the following on the interval 0 ≤ 𝑥 < 2𝜋 with exact values.
𝜋 3𝜋 5𝜋 7𝜋
𝑥= , ,
,
𝟏
4 4 4 4
_________________
4. 2 sin 𝑥 = csc 𝑥
2 sin 𝑥 =
𝐬𝐢𝐧 𝒙
2 sin2 𝑥 = 1
1
sin2 𝑥 = 2
sin 𝑥 = ±
√2
2
𝜋 3𝜋
𝑥= ,
2 2
_________________
5.
1 − 3 cos 𝑥 = sin2 𝑥
𝜋 5𝜋
𝑥= ,
3 3
_________________ 6.
1 − 3 cos 𝑥 = 𝟏 − 𝐜𝐨𝐬𝟐 𝒙
0 = 2 cos 2 𝑥
0 = cos 2 𝑥
tan2 𝑥 = 2 sec 𝑥 − 1
𝐬𝐞𝐜 𝟐 𝒙 − 𝟏 = 2 sec 𝑥 − 1
sec 2 𝑥 − 2 sec 𝑥 = 0
sec 𝑥 (sec 𝑥 − 2) = 0
sec 𝑥 = 0 or sec 𝑥 = 2
𝜋 𝜋 5𝜋
𝑥= , ,
2 6 6
_________________
7.
cos 𝑥 = ∅ or cos 𝑥 =
cot 2 𝑥 = 3(csc 𝑥 − 1)
1
2
csc 2 𝑥 − 1 = 3(csc 𝑥 − 1)
csc 2 𝑥 − 1 = 3 csc 𝑥 − 3
csc 2 𝑥 − 3 csc 𝑥 + 2 = 0
(csc 𝑥 − 1)(csc 𝑥 − 2) = 0
csc 𝑥 = 1 or csc 𝑥 = 2
1
sin 𝑥 = 1 or sin 𝑥 = 2
𝜋 7𝜋 11𝜋
𝑥= ,
,
2 6 6
_________________
8.
2 cos 2 𝑥 + sin 𝑥 = 1
2(𝟏 − 𝐬𝐢𝐧𝟐 𝒙) + sin 𝑥 = 1
2 − 2 sin2 𝑥 + sin 𝑥 = 1
2 sin2 𝑥 − sin 𝑥 − 1 = 0
(2 sin 𝑥 + 1)(sin 𝑥 − 1) = 0
1
𝜋 3𝜋 𝜋 11𝜋
𝑥= , , ,
2 2 6 6
_________________
9.
sin 𝑥 = − 2 or sin 𝑥 = 1
2 cos 2 𝑥 − √3 cos 𝑥 = 0
cos 𝑥 (2 cos 𝑥 − √3) = 0
cos 𝑥 = 0 or cos 𝑥 =
𝜋 5𝜋
𝑥= ,
3 3
_________________
10. cos 2𝑥 + 3 cos 𝑥 − 1 = 0
√3
2
(𝟐 𝐜𝐨𝐬𝟐 𝒙 − 𝟏) + 3 cos 𝑥 − 1 = 0
2 cos 2 𝑥 + 3 cos 𝑥 − 2 = 0
(2 cos 𝑥 − 1)(cos 𝑥 + 2) = 0
1
cos 𝑥 =
2𝜋 4𝜋
𝑥 = 0, 𝜋,
,
3 3
_________________
11. sin 2𝑥 = − sin 𝑥
or cos 𝑥 = −2
2
2 sin 𝑥 cos 𝑥 = − sin 𝑥
2 sin 𝑥 cos 𝑥 + sin 𝑥 = 0
sin 𝑥 (2 cos 𝑥 + 1) = 0
1
sin 𝑥 = 0 or cos 𝑥 = − 2
𝜋 5𝜋 7𝜋 11𝜋
𝑥= ,
, ,
6 6 6 6
_________________
12. tan 2𝑥 = cot 𝑥
𝟐𝐭𝐚𝐧 𝒙
𝟏
𝟏−𝐭𝐚𝐧𝟐 𝒙
= 𝐭𝐚𝐧 𝒙 cross mult.
2 tan2 𝑥 = 1 − tan2 𝑥
3 tan2 𝑥 = 1
1
tan2 𝑥 =
3
1
tan 𝑥 = ±√3 = ±
√3
3
Consider the period and/or frequency of the graphs when solving each of the following on the interval
0 ≤ 𝑥 < 2𝜋 with exact values.
𝜋 5𝜋 13𝜋 17𝜋
𝑥=
,
,
,
1
𝜋 5𝜋
2𝜋
12𝜋
12 12 12 12 13. sin 2𝑥 =
_________________
2𝑥 = 6 , 6
𝑇 = 𝟐 = 12
2
𝜋
5𝜋
𝑥 = 12 , 12 then add
𝜋 7𝜋 9𝜋 15𝜋
𝑥=
, ,
,
,
12 12 12 12 14. cos 3𝑥 = √2
_________________
2
17𝜋 23𝜋
,
12 12
3𝑥 = 4 ,
4𝜋 5𝜋 10𝜋 11𝜋
𝑥=
,
,
,
,
√3
9 9 9
9
_________________
15. sin 3𝑥 = − 2
16𝜋 17𝜋
,
9
9
3𝑥 =
12𝜋
12
𝜋 7𝜋
𝜋
to each till you reach 2𝜋
𝑇=
4
7𝜋
2𝜋
𝟑
=
8𝜋
12
8𝜋
𝑥 = 12 , 12 then add 12 to each till you reach 2𝜋
𝑥=
4𝜋 5𝜋
3
,
3
4𝜋 5𝜋
9
,
9
𝑇=
then add
6𝜋
9
2𝜋
𝟑
=
6𝜋
9
to each till you reach 2𝜋