Question

Solve 4cos^2x - 4sinx-1=0 (with steps)

Answer 1

Explanation :Simplifying

4cos²x + -4sinx + -1 =0

Reorder the terms:

-1 + 4cos²x + -4insx = 0

Solving

-1 + 4cos²x + -4insx = 0

Solving for variable 'c'.

Move all terms containing c to the left, all other terms to the right.

Add '1' to each side of the equation.

-1 + 4cos²x + 1 + -4insx = 0 + 1

Reorder the terms:

-1 + 1 + 4cos²x + -4insx = 0 + 1

Combine like terms: -1 + 1 = 0

0 + 4cos2x + -4insx = 0 + 1

4cos²x + -4insx = 0 + 1

Combine like terms: 0 + 1 = 1

4cos²x + -4insx = 1

Add '4insx' to each side of the equation.

4cos2x + -4insx + 4insx = 1 + 4insx

Combine like terms: -4insx + 4insx = 0

4cos²x + 0 = 1 + 4insx

4cos²x = 1 + 4insx

Divide each side by '4os2x'.

c = 0.25o-1s-2x-1 + ino-1s-1

Simplifying

c = 0.25o-1s-2x-1 + ino-1s-1

Reorder the terms:

c = ino-1s-1 + 0.25o-1s²x-1

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Answer 2

Answer:

The value of x is 30°.

Step-by-step explanation:

Given:-

The trigonometric function is 4cos²x - 4sinx - 1 = 0.

To find:-

The value of x.

As we know,

cos²x + sin²x = 1

⇒ cos²x = 1 - sin²x

Consider the given trigonometric function as follows:

4cos²x - 4sinx - 1 = 0

Substitute the value of cos²x as follows:

4($1$ - sin²x) - $4$sinx - 1 = 0

Simplify as follows:

4 - $4$sin²x - 4sinx - 1 = 0

-4sin²x - 4sinx + 3 = 0

4sin²x $+$ 4sinx - 3 = 0 _____ (1)

Now,

Let sinx = y, then equation (1) becomes,

4y² + 4y - 3 = 0

Using the middle-term splitting method, we have

4y² + 6y - 2y - 3 = 0

2y(2y + 3) - 1(2y + 3) = 0

(2y - 1)(2y + 3) = 0

y = 1/2 and y = -3/2.

Case1. When y = 1/2. Then,

sinx = 1/2

sinx = sin30°

x = 30°

Case2. When y = -3/2. Then,

sinx = -3/2

As we know the value of sine lies between -1 and 1.

Thus, sinx = -3/2 is not possible.

Therefore, the value of x is 30°.

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