Question

Cos (п-x)-sin (п/2 +x) =1

Answer 1

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Answer 2

To solve the trigonometric equation cos(π - x) - sin(π/2 + x) = 1, we'll first simplify the equation using trigonometric identities and then find the values of x that satisfy the equation.

Step 1: Apply trigonometric identities

cos(π - x) = -cos(x) (Identity: cos(π - θ) = -cos(θ)) sin(π/2 + x) = cos(x) (Identity: sin(π/2 + θ) = cos(θ))

Now, the equation becomes:

-cos(x) - cos(x) = 1

Step 2: Combine like terms

-2cos(x) = 1

Step 3: Isolate cos(x)

cos(x) = -1/2

Step 4: Find the solutions for x

To find the solutions for x, we need to determine the angles whose cosine value is equal to -1/2. Remember that cosine is negative in the second and third quadrants.

One common angle that has a cosine of -1/2 is 2π/3 (120 degrees) in the second quadrant. To find another angle in the third quadrant, we can use the symmetry of the unit circle.

In the third quadrant, the reference angle with a cosine of 1/2 is π - 2π/3 = π/3 (60 degrees). Therefore, the angle in the third quadrant with a cosine of -1/2 is π + π/3 = 4π/3 (240 degrees).

So, the solutions for x are:

x = 2π/3 + 2πn (where n is an integer) x = 4π/3 + 2πn (where n is an integer)

These are the general solutions to the equation. Plug in different integer values of n to find specific solutions for x.