Solve trigonometry equations

[hoosie]

Solve: 3sin(2X) + 4cos(2X) = 0
3sin(2X) = -4cos(2X)
Dividing both sides by cos(2X)
3sin(2X)/cos(2X) = -4cos(2X)/cos(2X)
3tan(2X) = -4
tan(2X) = -4/3

Consider a 3,4,5 right- angled triangle with reference angle θ, adjacent side 3, opposite side 4 and hypotenuse 5.
sinθ = O/H => sinθ = 4/5
cosθ = A/H => cosθ = 3/5
tanθ = O/A => tanθ = 4/3
θ = 0.92729522<sup>c

Solve tan(2X) = -4/3
Tangent function is negative in second and fourth quadrants.

First quadrant:
tan(2X) = 4/3
2X = tan-1(4/3)
= 0.92729522c (reference angle)

Second quadrant
2X = π - 0.92729522c
Fourth quadrant
2X = 2π - 0.92729522c

The period of tan(2X) is π/2 so values will repeat every π/2 in both directions.
X = (π/2 - 0.46364761 )c±πn/2
= (1.57079633 - 0.46364761)c ±πn/2
= 1.107149c ±πn/2
or
X = (π - 0.46364761)c ±πn/2
= (3.14159265 - 0.46364761)c ±πn/2
= 2.677945c ±πn/2

X = 1.107149c ±πn/2 or
X = 2.677945c ±πn/2
where n is a natural number.

Since 1.107149 + π/2
= 1.107149 + 1.570796
= 2.677945

the answer can be reduced to:

X = 1.107149c ±πn/2 where n is a natural number.

Use your graphics calculator to graph
f(X) = 3sin(2X) + 4cos(2X)</sup>

The X intercepts of this curve represent the solutions to 3sin(2X) + 4cos(2X) = 0
X intercepts are:
-6.7468, -5.1760, -3.6052, -2.0344,-0.4636,
1.1071, 2.6779, 4.2487, 5.8195, 7.3903
These values correspond to
X = 1.107149^c ±πn/2
= 1.107149 ± 1.570796n
where n is a natural number:

n = 5: -5.176035 - 1.570796 = -6.746831
n = 4: -3.605239 - 1.570796 = -5.176035
n = 3: -2.034443 - 1.570796 = -3.605239
n = 2: -0.463647 -1.570796 = -2.034443
n = 1: 1.107149 - 1.570796 = -0.463647
n = 0: 1.107149
n = 1: 1.107149+ 1.570796 = 2.677945
n = 2: 2.677945 + 1.570796 = 4.248741
n = 3: 4.248741 + 1.570796 = 5.819537
n = 4: 5.819537 + 1.570796 = 7.390333

 

[Otis]

I see placeholders instead of images.