Question

Cos2x+cos^2(x-pi/2)=0,75

Answer 1

Ответ:

Cos^2(x-90)=sin^x

Cos2x=cos^2x-sin^2x

cos^2x-sin^2x+sin^2x=3/4

cosx=+-sqrt(3)/2

X=+-p/6+2pk

Answer 2

To solve the equation cos(2x) + cos^2(x - π/2) = 0.75, we'll first manipulate the equation to simplify it and then find the solutions for x.

Step 1: Rewrite the equation in terms of trigonometric identities. Recall the double-angle formula for cosine: cos(2x) = 2cos^2(x) - 1

Also, recall the identity: cos(x - π/2) = -sin(x)

Substitute these identities into the equation: 2cos^2(x) - 1 + (-sin^2(x)) = 0.75

Step 2: Combine like terms: 2cos^2(x) - sin^2(x) - 1 = 0.75

Step 3: Simplify the equation: 2cos^2(x) - sin^2(x) = 0.75 + 1

Step 4: Further simplify: 2cos^2(x) - sin^2(x) = 1.75

Step 5: Now, recall the Pythagorean identity: sin^2(x) + cos^2(x) = 1 Rearrange it to express sin^2(x) in terms of cos^2(x): sin^2(x) = 1 - cos^2(x)

Step 6: Substitute the value of sin^2(x) in terms of cos^2(x) into the equation: 2cos^2(x) - (1 - cos^2(x)) = 1.75

Step 7: Solve for cos^2(x): 2cos^2(x) - 1 + cos^2(x) = 1.75 3cos^2(x) - 1 = 1.75 3cos^2(x) = 1.75 + 1 3cos^2(x) = 2.75

Step 8: Isolate cos^2(x): cos^2(x) = 2.75 / 3 cos^2(x) ≈ 0.9167

Step 9: Take the square root of both sides to find the value of cos(x): cos(x) ≈ ±√(0.9167) cos(x) ≈ ±0.957

Step 10: Now, find the possible values of x by taking the inverse cosine (cos^(-1)) of each value: x ≈ ±cos^(-1)(0.957)

Using a calculator, you can find the approximate values of x: x ≈ ±16.64° (rounded to two decimal places)

Keep in mind that trigonometric functions are periodic, and there are infinitely many solutions to this equation due to the periodic nature of the cosine function. The general solution for x is:

x ≈ 16.64° + k * 360° and x ≈ -16.64° + k * 360°

where k is an integer, allowing for multiple solutions as you move around the unit circle.