E-Z HL Trig Problems
1. Simplify the following expressions:
(a) cos 37° cos 23° − sin 37°sin 23°
cos 37°cos 23° − sin 37°sin 23°
= cos ( 37° + 23° )
= cos ( 60° )
=
1
2
⎛π
⎞
⎛π
⎞
(c) sin ⎜ + x ⎟ − sin ⎜ − x ⎟
⎝3
⎠
⎝3
⎠
(b) sin 28° cos 42° + sin 28° cos 42°
sin 28°cos 42° + sin 28°cos 42°
= sin ( 28° + 42° )
= sin 70°
⎛π
⎞
⎛π
⎞
(d) cos ⎜ + x ⎟ + cos ⎜ − x ⎟
⎝6
⎠
⎝6
⎠
⎛π
⎞
⎛π
⎞
sin ⎜ + x ⎟ − sin ⎜ − x ⎟
⎝3
⎠
⎝3
⎠
⎛π
⎞
⎛π
⎞
cos ⎜ + x ⎟ + cos ⎜ − x ⎟
⎝6
⎠
⎝6
⎠
π
π
π
π
cos x + cos sin x − sin cos x + cos sin x
3
3
3
3
π
= 2 cos sin x
3
1
= 2 sin x
2
= sin x
π
π
π
π
= cos cos x − sin sin x + cos cos x + sin sin x
6
6
6
6
π
= 2 cos cos x
6
3
=2
cos x
2
= 3 cos x
= sin
π⎞
⎛ 2π
⎞
⎛
− x ⎟ + sin ⎜ x − ⎟
(e) sin ⎜
⎝ 3
⎠
⎝
3⎠
π⎞
⎛ 2π
⎞
⎛
sin ⎜
− x ⎟ + sin ⎜ x − ⎟
⎝ 3
⎠
⎝
3⎠
2π
2π
π
π
cos x − cos
sin x + sin x cos − cos x sin
3
3
3
3
3
1
1
3
=
cos x + sin x + sin x −
cos x
2
2
2
2
= sin x
= sin
sin θ + sin 3θ + sin 5θ
= tan 3θ
2. Prove: cosθ + cos 3θ + cos 5θ
sin θ + sin 3θ + sin 5θ
cosθ + cos 3θ + cos 5θ
sin(3θ − 2θ ) + sin 3θ + sin(3θ + 2θ )
=
cos(3θ − 2θ ) + cos 3θ + cos(3θ + 2θ )
sin 3θ cos 2θ − cos 3θ sin 2θ + sin 3θ + sin 3θ cos 2θ + cos 3θ sin 2θ
=
cos 3θ cos 2θ + sin 3θ sin 2θ + cos 3θ + cos 3θ cos 2θ − sin 3θ sin 2θ
2 sin 3θ cos 2θ + sin 3θ
=
2 cos 3θ cos 2θ + cos 3θ
sin 3θ (2 cos 2θ + 1)
=
cos 3θ (2 cos 2θ + 1)
sin 3θ
=
cos 3θ
= tan 3θ
3. Prove:
sin(x − y) + sin x + sin(x + y)
= tan x
cos(x − y) + cos x + cos(x + y)
sin(x − y) + sin x + sin(x + y)
cos(x − y) + cos x + cos(x + y)
sin x cos y − cos x sin y + sin x + sin x cos y + cos x sin y
=
cos x cos y + sin x sin y + cos x + cos x cos y − sin x sin y
2 sin x cos y + sin x
=
2 cos x cos y + cos x
sin x(2 cos y + 1)
=
cos x(2 cos y + 1)
sin x
=
cos x
= tan x
4. Prove: cos(A + B + C) + cos(A + B − C) + cos(A − B + C) + cos(−A + B + C) = 4 cos A cos B cosC
cos(A + B + C) + cos(A + B − C) + cos(A − B + C) + cos(−A + B + C)
= cos A cos(B + C) − sin Asin(B + C) + cos A cos(B − C) − sin Asin(B − C) + cos A cos(B − C) + sin Asin(B − C) + cos(−A)cos(B + C) − sin(−A)sin(B + C)
= cos A cos(B + C) + cos A cos(B − C) − sin Asin(B − C) + cos A cos(B − C) + sin Asin(B − C) + cos(A)cos(B + C)
= 2 cos A cos(B + C) + 2 cos A cos(B − C)
= 2 cos A cos B cosC − 2 cos Asin Bsin C + 2 cos A cos B cosC + 2 cos Asin Bsin C
= 4 cos A cos B cosC
5. Solve the equation 4 tan 2 x + 12 sec x + 1 = 0 giving all solutions in degrees, to the nearest degree, in the interval −180° < x < 180°.
4 tan 2 x + 12 sec x + 1 = 0
(
)
4 sec 2 x − 1 + 12 sec x + 1 = 0
4 sec 2 x + 12 sec x − 3 = 0
−12 ± 144 + 48
8
−12 ± 8 3
sec x =
8
−3 ± 2 3
sec x =
2
2
cos x =
−3 ± 2 3
x  ±108°
sec x =
6. Solve the equation
3 tan θ − sec θ = 1 , giving all solutions in the interval 0° < θ < 360°.
3 tan θ − sec θ = 1
(
3 sin θ
1
−
=1
cosθ
cosθ
3 sin θ − 1
=1
cosθ
)
3 sin θ − 1 = ( cosθ )
2
2
3sin 2 θ − 2 3 sin θ + 1 = cos 2 θ
3sin 2 θ − 2 3 sin θ + 1 = 1 − sin 2 θ
4 sin 2 θ − 2 3 sin θ = 0
(
)
2 sin θ 2 sin θ − 3 = 0
sin θ = 0 → θ = 0°,180°, 360°
3
→ θ = 60°,120°
2
since we squared the equation check all answers
sin θ =
θ = 0° ,180°, 360° , 60°, 120°
7. Prove the identity sin 3A = 3sin A − 4 sin 3 A . Hence show that sin10° is a root of the equation 8x 3 − 6x + 1 = 0 .
sin 30° = sin 3(10°)
sin 3A = 3sin A − 4 sin 3 A
1
= sin 2A cos A + cos 2Asin A
= 3sin10° − 4 sin 3 10°
2
= 2 sin A cos A cos A + (1 − 2 sin 2 A)sin A
Let
sin10° = x
= 2 sin A cos 2 A + sin A − 2 sin 3 A
1
= 3x − 4x 3
= 2 sin A(1 − sin 2 A) + sin A − 2 sin 3 A
2
3
3
= 2 sin A − 2 sin A + sin A − 2 sin A
1 = 6x − 8x 3
= 3sin A − 4 sin 3 A
8x 3 − 6x + 1 = 0
8. Prove the identity tan θ + cot θ = 2 csc 2θ . Find in radians, all the solutions of the equation tan x + cot x = 8 cos 2x in the interval 0 < x < π .
tan θ + cot θ
sin θ cosθ
=
+
cosθ sin θ
sin 2 θ + cos 2 θ
=
sin θ cosθ
1
=
sin θ cosθ
2
=
2 sin θ cosθ
2
=
sin 2θ
= 2 csc 2θ
tan x + cot x = 8 cos 2x
2 csc 2x = 8 cos 2x
2
= 8 cos 2x
sin 2x
2 = 8 sin 2x cos 2x
2 = 4 × 2 sin 2x cos 2x
1
= sin 4x
2
π 5π 13π 17π
4x = , ,
,
6 6 6
6
π 5π 13π 17π
x=
, ,
,
24 24 24 24
9. A triangle has sides of length 2cm, 2cm and 4cm. Find the area of the triangle.
Area = 0 since 2+2=4 it is one dimensional
10. Given that a=27m, b=32m and c=30.6m in triangle ABC, find the smallest angle of the triangle, and its area.
32 2 + 30.6 2 − 27 2
 0.629
2(32)(30.6)
θ  51.0°
1
A = (32)(30.6)sin 51.0°
2
A = 381m 2
cosθ =
11. Solve 2 sin 2 x + 5 cos x + 1 = 0, −π ≤ x ≤ π
2 sin 2 x + 5 cos x + 1 = 0
2(1 − cos 2 x) + 5 cos x + 1 = 0
−2 cos 2 x + 5 cos x + 3 = 0
2 cos 2 x − 5 cos x − 3 = 0
(2 cos x + 1)(cos− 3) = 0
−1
cos x = , cos x = 3
2
2π −2π
x=
,
, ( cos x = 3 → no solution )
3
3
12. Solve 2 sin 2x = tan x, 0° ≤ x ≤ 180°
2 sin 2x = tan x
sin x
4 sin x cos x =
cos x
2
4 sin x cos x = sin x
(
)
4 sin x 1 − sin 2 x − sin x = 0
(
)
sin x ( 3 − 4 sin x ) = 0
sin x 4 − 4 sin 2 x − 1 = 0
2
± 3
2
x = 0°,180°, 60°,120°
sin x = 0,sin x =
13. Solve 2 csc x = 5 cot x, 0° ≤ x ≤ 180°
2 csc x = 5 cot x
2
5 cos x
=
sin x
sin x
2 sin x = 5 sin x cos x
2 sin x − 5 sin x cos x = 0
sin x ( 2 − 5 cos x ) = 0
2
5
x = 0°,180°; x  66.4°
sin x = 0, cos x =
14. Solve 3cos x = 2 cos 2x, 0° ≤ x ≤ 360°
3cos x = 2 cos 2x
3cos x = 2(2 cos 2 x − 1)
3cos x = 4 cos 2 x − 2
4 cos 2 x − 3cos x − 2 = 0
x  115°, 245°
1
15. Solve sin x + sin 2x = 0, 0 ≤ x ≤ 2π
2
1
sin x + sin 2x = 0
2
sin x + sin x cos x = 0
sin x (1 + cos x ) = 0
sin x = 0, cos x = −1
x = 0, π , 2π
16. Solve 6 cos x − 1 = sec x, 0° ≤ x ≤ 180°
6 cos x − 1 = sec x
1
6 cos x − 1 =
cos x
2
6 cos x − cos x − 1 = 0
(2 cos x − 1)(3cos x + 1) = 0
1
−1
cos x = , cos x =
2
3
x = 60°, x  109°
17. Solve tan 2 x + 3sec x = 0, 0 ≤ x ≤ 2π
tan 2 x + 3sec x = 0
(sec
2
)
x − 1 + 3sec x = 0
sec 2 x + 3sec x − 1 = 0
1 + 3cos x − cos 2 x = 0
cos 2 x − 3cos x − 1 = 0
x  1.88, 4.40
18. Solve 6 tan 2 x = 4 sin 2 x + 1, 0° ≤ x ≤ 360°
6 tan 2 x = 4 sin 2 x + 1
6 sin 2 x
= 4 sin 2 x + 1
2
cos x
6 sin 2 x = 4 sin 2 x + 1 1 − sin 2 x
(
)(
)
6 sin x = 4 sin x + 1 − 4 sin x − sin 2 x
2
2
4
4 sin 4 x + 3sin 2 x − 1 = 0
( 4 sin
2
)(
)
x − 1 sin 2 x + 1 = 0
1
sin x = ± ,sin x = −1
2
x = 30°,150°, 270°
19. Solve 3cot 2 x + 5 csc x + 1 = 0, 0 ≤ x ≤ 2π
3cot 2 x + 5 csc x + 1 = 0
3cos 2 x
5
+
+1= 0
2
sin x
sin x
3(1 − sin 2 x) + 5 sin x + sin 2 x = 0
−2 sin 2 x + 5 sin x + 3 = 0
2 sin 2 x − 5 sin x − 3 = 0
(2 sin x + 1)(sin x − 3) = 0
−1
sin x = ,sin x = 3
2
7π 11π
x=
,
, ( sin x = 3 → no solution )
6 6
20. Solve csc x = 3 sec 2 x, 0 ≤ x ≤ π
csc x = 3 sec 2 x
1
3
=
sin x cos 2 x
1 − sin 2 x = 3 sin x
sin 2 x + 3 sin x − 1 = 0
No solution within domain
21. Solve sec 4 x + 2 = 6 tan 2 x, 0° ≤ x ≤ 180°
sec 4 x + 2 = 6 tan 2 x
(
)
sec 4 x + 2 = 6 sec 2 x − 1
sec 4 x − 6 sec 2 x + 8 = 0
(sec
2
)(
)
x − 4 sec 2 x − 2 = 0
sec x = ±2,sec x = 2
1
1
cos x = ± , cos =
2
2
x = 60°,120°, 45°
22. Solve cos 2x cos x = sin 2x sin x, −180° ≤ x ≤ 180°
cos 2x cos x = sin 2x sin x
( 2 cos x − 1) cos x = 2 sin x cos x
( 2 cos x − 1) cos x = 2 (1 − cos x ) cos x
( 2 cos x − 1) cos x − 2 (1 − cos x ) cos x = 0
cos x ( 2 cos x − 1 − 2 + 2 cos x ) = 0
cos x ( 4 cos x − 3) = 0
2
2
2
2
2
2
2
2
3
2
x = 90°, −90°, ±30°, ±150°
cos x = 0, cos x = ±
2