Solve the equation 2sin(2x) = cos x.
and
Solve the equation
cos^2(2x)–sin^2(2x)= (√3 / 2).
1st one:
2sin(2x) = cosx
4sinxcosx - cosx = 0
cosx(4sinx - 1) = 0
cosx = 0 or sinx = 1/4
for cosx = 0
x = 90° or 270° OR x = π/2 or 3π/2 radians
for sinx=1/4
x = appr 14.5° or 165.5° OR appr .253 or 2.89 radians
2nd:
cos^2(2x)–sin^2(2x)= (√3 / 2)
cos (4x) = √3/2 , using cos (2A) = cos^2 A - sin^2 A
I know cos 30° = cos 330° = √3/2
so 4x = 30° or 4x = 330°
x = 7.5° or x = 82.5°
the period of cos 4x is 360/4 ° = 90°
so adding or subtracting multiples of 90 to any of the above solutions, will yield a new solution
e.g. 7.5 + 90 = 97.5°
97.5+90 = 187.5° etc
check for x = 187.5°
LS = cos^2 375° - sin^2 375°
= .9330... - .0669...
= .866025...
RS = √3/2 = .866025...
= LS
(set your calculator to Radians using the DRG key, and get the equivalent answers in radians)
To solve the equation 2sin(2x) = cos x, we can use the following steps:
Step 1: Simplify the equation using trigonometric identities.
We know that sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x). Substituting these identities into the equation, we get:
2(2sin(x)cos(x)) = cos(x)
Step 2: Simplify further.
4sin(x)cos(x) = cos(x)
Step 3: Move all terms to one side of the equation.
4sin(x)cos(x) - cos(x) = 0
Step 4: Factor out common terms.
cos(x)(4sin(x) - 1) = 0
Step 5: Set each factor equal to zero and solve.
cos(x) = 0 (Solution 1)
4sin(x) - 1 = 0
4sin(x) = 1
sin(x) = 1/4 (Solution 2)
For the equation cos^2(2x) – sin^2(2x) = (√3 / 2), we can use the following steps:
Step 1: Simplify the equation.
We know that cos^2(2x) = (cos(2x))^2 and sin^2(2x) = (sin(2x))^2. Substituting these into the equation, we get:
(cos(2x))^2 – (sin(2x))^2 = (√3 / 2)
Step 2: Use the Pythagorean identity.
(cos(2x))^2 – (1 - (cos(2x))^2) = (√3 / 2)
Step 3: Simplify further.
2(cos(2x))^2 – 1 = (√3 / 2)
Step 4: Add 1 to both sides of the equation.
2(cos(2x))^2 = (√3 / 2) + 1
Step 5: Simplify and solve for cos(2x).
2(cos(2x))^2 = (√3 + 2) / 2
(cos(2x))^2 = (√3 + 2) / 4
Step 6: Take the square root of both sides.
cos(2x) = ±√[(√3 + 2) / 4]
Step 7: Solve for x.
To find the values of x, we need to solve for the values of 2x.
2x = ±arccos(√[(√3 + 2) / 4])
Step 8: Divide both sides by 2 and solve for x.
x = ±(1/2)arccos(√[(√3 + 2) / 4])
To solve the equation 2sin(2x) = cos(x), we can use trigonometric identities to simplify the equation and solve for x.
1. Start by using the double-angle identity for sine: sin(2x) = 2sin(x)cos(x).
This transforms the equation to: 2 * 2sin(x)cos(x) = cos(x).
2. Simplify the equation further: 4sin(x)cos(x) = cos(x).
3. Rearrange the terms: 4sin(x)cos(x) - cos(x) = 0.
4. Factor out the common term: cos(x)(4sin(x) - 1) = 0.
5. Set each factor equal to zero:
a) cos(x) = 0
b) 4sin(x) - 1 = 0
6. Solve for x in each equation:
a) cos(x) = 0:
The cosine function equals zero at 90-degree intervals.
So, x = 90 degrees or x = 270 degrees.
b) 4sin(x) - 1 = 0:
Add 1 to both sides: 4sin(x) = 1.
Divide both sides by 4: sin(x) = 1/4.
Take the inverse sin (sin^(-1)) of both sides: x = sin^(-1)(1/4).
Therefore, the solutions for the equation 2sin(2x) = cos(x) are:
- x = 90 degrees, 270 degrees, and the inverse sine of 1/4.
To solve the equation cos^2(2x) - sin^2(2x) = (√3 / 2), we can again use trigonometric identities to simplify the equation and find the solutions.
1. Use the Pythagorean identity: cos^2(x) + sin^2(x) = 1.
Square both sides of the equation: [cos^2(2x) - sin^2(2x)]^2 = (√3 / 2)^2.
Simplify: cos^4(2x) - 2cos^2(2x)sin^2(2x) + sin^4(2x) = 3/4.
2. We can rewrite cos^2(2x) as 1 - sin^2(2x) using the Pythagorean identity.
Substitute: (1 - sin^2(2x)) - sin^2(2x) + sin^4(2x) = 3/4.
Simplify: 1 - 2sin^2(2x) + sin^4(2x) = 3/4.
3. Rearrange the equation: sin^4(2x) - 2sin^2(2x) + 1 - 3/4 = 0.
Combine like terms: sin^4(2x) - 2sin^2(2x) + 1/4 = 0.
4. Factor the quadratic expression: (sin^2(2x) - 1/2)^2 = 0.
5. Take the square root of both sides: sin^2(2x) - 1/2 = 0.
Solve for sin(2x): sin^2(2x) = 1/2.
Take the square root again: sin(2x) = ±√(1/2).
6. Solve for x:
a) sin(2x) = √(1/2):
Take the inverse sin: 2x = sin^(-1)(√(1/2)).
Solve for x: x = (1/2) * sin^(-1)(√(1/2)).
b) sin(2x) = -√(1/2):
Take the inverse sin: 2x = sin^(-1)(-√(1/2)).
Solve for x: x = (1/2) * sin^(-1)(-√(1/2)).
Therefore, the solutions for the equation cos^2(2x) - sin^2(2x) = (√3 / 2):
- x = (1/2) * sin^(-1)(√(1/2)) and x = (1/2) * sin^(-1)(-√(1/2)).