Solving Trigonometric Equations General Solution Of 7 Cos X - 2 Sin^2 X + 5 = 0

Delving into the realm of trigonometric equations, we embark on a journey to decipher the general solution for the equation 7 cos x - 2 sin^2 x + 5 = 0. This equation, a blend of cosine and sine functions, presents an intriguing challenge that demands a strategic approach. Our exploration will involve transforming the equation into a more manageable form, employing trigonometric identities, and ultimately unraveling the infinite set of solutions that satisfy this mathematical puzzle.

Transforming the Equation: A Strategic Maneuver

The key to unlocking the solution lies in recognizing the inherent relationship between sine and cosine functions. The Pythagorean identity, a cornerstone of trigonometry, states that sin^2 x + cos^2 x = 1. This identity serves as our bridge, allowing us to express sin^2 x in terms of cos^2 x, or vice versa. In this case, we choose to replace sin^2 x with (1 - cos^2 x), transforming our equation into a quadratic form solely in terms of cosine.

Substituting (1 - cos^2 x) for sin^2 x, our equation metamorphoses into:

7 cos x - 2(1 - cos^2 x) + 5 = 0

Expanding and rearranging the terms, we arrive at a quadratic equation in cos x:

2 cos^2 x + 7 cos x + 3 = 0

This quadratic form is a significant step forward, as it allows us to employ familiar algebraic techniques to solve for cos x. The equation now resembles a standard quadratic equation of the form ax^2 + bx + c = 0, where our variable is cos x.

Solving the Quadratic Equation: Unveiling the Cosine Values

With our equation now in quadratic form, we turn to the task of finding the values of cos x that satisfy it. We have several options at our disposal, including factoring, completing the square, or employing the quadratic formula. In this case, factoring proves to be the most efficient route.

We seek two numbers that multiply to (2 * 3 = 6) and add up to 7. These numbers are 6 and 1. Using these numbers, we can rewrite the middle term and factor the quadratic expression:

2 cos^2 x + 6 cos x + cos x + 3 = 0

Factoring by grouping, we obtain:

2 cos x (cos x + 3) + 1 (cos x + 3) = 0

(2 cos x + 1)(cos x + 3) = 0

This factored form reveals the two possible values for cos x that satisfy the equation:

2 cos x + 1 = 0 or cos x + 3 = 0

Solving these equations, we find:

cos x = -1/2 or cos x = -3

However, we must recognize that the cosine function has a range of [-1, 1]. This means that the solution cos x = -3 is extraneous and must be discarded. Therefore, the only valid cosine value is:

cos x = -1/2

Finding the Principal Solutions: Pinpointing the Angles

Now that we have determined the cosine value, our next task is to find the angles x that satisfy cos x = -1/2. We need to identify the principal solutions, which are the angles within the interval [0, 2π) that have a cosine of -1/2.

Recall the unit circle and the definition of cosine as the x-coordinate of a point on the unit circle. We seek angles whose terminal side intersects the unit circle at a point with an x-coordinate of -1/2. These angles lie in the second and third quadrants.

The reference angle for cos x = 1/2 is π/3. Therefore, the angles in the second and third quadrants with a cosine of -1/2 are:

• x = π - π/3 = 2π/3
• x = π + π/3 = 4π/3

These are our principal solutions, the angles within one full revolution that satisfy the equation.

General Solution: Capturing All Possibilities

While the principal solutions provide us with specific angles, the general solution encompasses all possible angles that satisfy the equation. Since the cosine function is periodic with a period of 2π, we can add integer multiples of 2π to our principal solutions to generate an infinite set of solutions.

The general solution is expressed as:

• x = 2π/3 + 2πk, where k is an integer
• x = 4π/3 + 2πk, where k is an integer

These two expressions represent all possible angles that have a cosine of -1/2. The integer k allows us to cycle through all coterminal angles, capturing the periodic nature of the cosine function.

In Conclusion: A Symphony of Solutions

We have successfully navigated the trigonometric landscape, transforming, solving, and generalizing to arrive at the complete solution set for the equation 7 cos x - 2 sin^2 x + 5 = 0. By employing the Pythagorean identity, solving a quadratic equation, and understanding the periodicity of the cosine function, we have unveiled the infinite family of solutions represented by the general solution:

• x = 2π/3 + 2πk, where k is an integer
• x = 4π/3 + 2πk, where k is an integer

This journey exemplifies the power of trigonometric identities and algebraic techniques in unraveling the complexities of trigonometric equations.

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