Solving Trigonometric Equations: A Step-by-Step Guide

Hey there, algebra enthusiasts! Let's dive into solving some trigonometric equations. We'll break down each problem, step by step, to make sure you understand the process. We'll be looking for solutions within specific intervals and expressing our answers in degrees. Ready to get started? Let's go!

Solving 2cos(x/2) - √3 = 0 within (0°, 90°)

Alright, guys, let's tackle the first problem: 2cos(x/2) - √3 = 0 within the interval (0°, 90°). Our goal is to find the value(s) of x that satisfy this equation and fall within the given range. Here's how we'll do it:

1. Isolate the cosine function: First things first, let's isolate the cosine term. Add √3 to both sides of the equation: 2cos(x/2) = √3 Then, divide both sides by 2: cos(x/2) = √3 / 2

2. Find the reference angle: Now, we need to determine the angle whose cosine is √3 / 2. Recall that the cosine function represents the x-coordinate on the unit circle. The reference angle for which cos(θ) = √3 / 2 is 30°. You can remember this from your basic trigonometric values or by using a unit circle.

3. Find the general solutions: Since the cosine function is positive in the first and fourth quadrants, we need to find the angles in these quadrants that have a reference angle of 30°. However, we're dealing with x/2, so we need to account for this. The general solutions for x/2 are: x/2 = 30° + 360°k or x/2 = 330° + 360°k, where k is an integer.

4. Solve for x: Multiply both sides of each equation by 2 to solve for x: x = 60° + 720°k or x = 660° + 720°k

5. Find solutions within the given interval: We're looking for solutions in the interval (0°, 90°). Let's plug in different integer values for k and see which solutions fall within this range:

   • For k = 0:
     • x = 60° + 720°(0) = 60° (This is within our interval!)
     • x = 660° + 720°(0) = 660° (This is outside our interval)
   • For k = 1 and higher, the values will be far outside our interval. For k = -1, the values will be negative and outside the interval.

6. Final answer: The only solution within the interval (0°, 90°) is x = 60°.

So, there you have it, folks! We've successfully solved the first equation. Notice how we systematically isolated the trigonometric function, found the reference angle, determined general solutions, solved for x, and finally, identified the solution within the specified interval. Let's move on to the next problem!

Solving tan(2x + π/3) = 1 within (0, π/2)

Alright, let's tackle the equation tan(2x + π/3) = 1 within the interval (0, π/2). Here's how we'll break this one down:

1. Find the reference angle: We need to figure out the angle whose tangent is 1. Remember, the tangent function represents the ratio of the sine to the cosine (sin/cos). The reference angle for which tan(θ) = 1 is 45° or π/4 radians. Because the tangent function has a period of π, we know that tangent is also positive in the third quadrant.

2. Find the general solutions: We can express the general solutions as: 2x + π/3 = π/4 + kπ, where k is an integer.

3. Solve for x: Let's isolate x. First, subtract π/3 from both sides: 2x = π/4 - π/3 + kπ Simplify the right side by finding a common denominator (12): 2x = (3π - 4π) / 12 + kπ 2x = -π/12 + kπ Now, divide by 2: x = -π/24 + kπ/2

4. Find solutions within the given interval: We're looking for solutions in the interval (0, π/2). Let's plug in different integer values for k and see which solutions fall within this range:

   • For k = 0:
     • x = -π/24 + (0)π/2 = -π/24 (This is outside our interval)
   • For k = 1:
     • x = -π/24 + (1)π/2 = -π/24 + 12π/24 = 11π/24 (This is within our interval!)
   • For k = 2:
     • x = -π/24 + (2)π/2 = -π/24 + 24π/24 = 23π/24 (This is outside our interval)
   • For k values greater than 2 and less than 0, the values are outside our interval.

5. Final answer: The only solution within the interval (0, π/2) is x = 11π/24. If we convert to degrees, we get approximately 82.5°.

There you go! Another equation solved. This time, we worked with the tangent function. The key steps are consistent: identify the reference angle, find general solutions, solve for x, and check the solutions against the given interval. Let's keep the momentum going!

Solving 3cot(x/2 - π/6) = √3 within (-π, 0)

Alright, let's work through the final problem: 3cot(x/2 - π/6) = √3 within the interval (-π, 0). Here's our game plan:

1. Isolate the cotangent function: First, divide both sides by 3: cot(x/2 - π/6) = √3 / 3

2. Find the reference angle: The cotangent function is the reciprocal of the tangent function (cot = 1/tan). So, if cot(θ) = √3 / 3, then tan(θ) = 3 / √3 = √3. The angle whose tangent is √3 is 60° or π/3 radians. Remember that cotangent is positive in the first and third quadrants.

3. Find the general solutions: The general solutions can be expressed as: x/2 - π/6 = π/3 + kπ, where k is an integer.

4. Solve for x: Let's isolate x. First, add π/6 to both sides: x/2 = π/3 + π/6 + kπ Simplify the right side: x/2 = 2π/6 + π/6 + kπ x/2 = 3π/6 + kπ x/2 = π/2 + kπ Now, multiply by 2: x = π + 2kπ or x = (2k+1)π

5. Find solutions within the given interval: We're looking for solutions in the interval (-π, 0). Let's plug in different integer values for k and see which solutions fall within this range:

   • For k = -1:
     • x = π + 2(-1)π = π - 2π = -π (This is outside our interval)
   • For k = 0:
     • x = π + 2(0)π = π (This is outside our interval)
   • For k = -1:
     • x = (2(-1)+1)π= -π (This is outside our interval)
   • For k = -2:
     • x = (2(-2)+1)π= -3π (This is outside our interval)

We need to revisit our general solutions and double check our work since all of our solutions didn't fall within the interval. Going back to step 3.

3. Find the general solutions: The cotangent function has a period of π. The general solutions can be expressed as: x/2 - π/6 = π/3 + kπ, where k is an integer.

4. Solve for x: Let's isolate x. First, add π/6 to both sides: x/2 = π/3 + π/6 + kπ Simplify the right side: x/2 = 2π/6 + π/6 + kπ x/2 = 3π/6 + kπ x/2 = π/2 + kπ Now, multiply by 2: x = π + 2kπ

   or x = (2k+1)π

5. Find solutions within the given interval: We're looking for solutions in the interval (-π, 0). Let's plug in different integer values for k and see which solutions fall within this range:

If we let k = -1, we get x = π + 2(-1)π = π - 2π = -π. This is not in the range (-π, 0).

If we let k = 0, we get x = π + 2(0)π = π. This is not in the range (-π, 0).

If we let k = -1, we get x = (2(-1)+1)π = -π. This is not in the range (-π, 0).

There seems to be an error in the provided solutions. The cotangent has a period of pi, and its positive in the first and third quadrants. However, it seems that there are no solutions within this range.

6. Final answer: No solutions within the interval (-π, 0).

And that wraps up our final problem! We've successfully solved three trigonometric equations, each with its own unique characteristics. Remember, the key is to be methodical: isolate the trigonometric function, find the reference angle, determine the general solutions, solve for x, and finally, identify the solutions within the specified interval. Keep practicing, and you'll become a pro in no time! Keep up the great work and happy solving!