Question

А) 2cos(x+pi/2)=-1 б) sin(-2x)=1

Answer 1

N1

2cos(x + π/2) = -1

cos(x + π/2) = -1/2

[x + π/2 = 2π/3 + 2πn

[x + π/2 = -2π/3 + 2πn

[x = π/6 + 2πn

[x = -7π/6 + 2πn n € Z

Ответ : x = π/6 + 2πn ; x = -7π/6 + 2πn ; n € Z

N2

sin(-2x) = 1

-sin2x = 1

sin2x = -1

2x = -π/2 + 2πk

x = -π/4 + πk ; k € Z

Ответ : -π/4 + πk ; k € Z

Answer 2

Let's solve each trigonometric equation step by step:

a) 2cos(x + π/2) = -1

To solve this equation, we want to find the values of x that satisfy the equation. Recall that the cosine function has a period of 2π, which means its values repeat every 2π.

Step 1: Shift the angle using the cosine addition formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

In this case, A = x and B = π/2, so the equation becomes: 2(cos(x)cos(π/2) - sin(x)sin(π/2)) = -1

Step 2: Simplify using known values: 2(0 - sin(x)) = -1 -2sin(x) = -1

Step 3: Isolate sin(x): sin(x) = 1/2

Now, we need to find the solutions for sin(x) = 1/2. Since the sine function has a period of 2π as well, we look for solutions in the interval [0, 2π).

Step 4: Find the solutions for sin(x) = 1/2: x = π/6 and x = 5π/6

b) sin(-2x) = 1

Step 1: There is no simple trigonometric identity for sin(-2x), so we'll use the definition of sine for negative angles: sin(-θ) = -sin(θ)

The equation becomes: -sin(2x) = 1

Step 2: Isolate sin(2x): sin(2x) = -1

Step 3: Find the solutions for sin(2x) = -1. We look for solutions in the interval [0, 2π).

Step 4: Find the half-angle identity for sine: sin(2x) = 2sin(x)cos(x)

Step 5: Set up the equation: 2sin(x)cos(x) = -1

Step 6: Divide both sides by 2: sin(x)cos(x) = -1/2

Now, we need to find the solutions for sin(x)cos(x) = -1/2.

Step 7: We can approach this using the double-angle formula for sine: sin(2θ) = 2sin(θ)cos(θ)

Comparing this with sin(x)cos(x) = -1/2, we can see that we need to find angles x where sin(2x) = -1/2.

Step 8: Find the solutions for sin(2x) = -1/2:

To do this, we can use the inverse sine function (sin^(-1)): 2x = sin^(-1)(-1/2)

Step 9: Solve for x: x = (sin^(-1)(-1/2)) / 2

Use a calculator to find the numerical value of (sin^(-1)(-1/2)), and then divide it by 2 to find the solutions for x. Remember that we are looking for solutions in the interval [0, 2π).

Keep in mind that some angles may result in multiple solutions due to the periodic nature of trigonometric functions.