# Solving Trigonometric Equations By Finding All Solutions

TLDRThis educational video script explores trigonometric equations involving sine, cosine, and tangent functions. It demonstrates how to find the values of x for various equations, such as sine x = 1/2, cosine x = -√2/2, and more complex scenarios like 2sin(x) + √3 = 0. The script explains the use of reference angles and the importance of considering all quadrants to find comprehensive solutions. It also covers how to generalize solutions using n for sine and cosine with a period of 2π, and for tangent with a period of π. The script concludes with simplifying the solution for sin^2(x) - 1 = 0, emphasizing the significance of understanding periodicity and the unit circle in trigonometry.

###### Takeaways

- 📚 The script explains how to find the values of x for which sine x equals one-half using the unit circle, identifying angles at pi/6 and 5pi/6.
- 🔍 To find all solutions, the script suggests adding multiples of 2pi to the initial solutions, resulting in the general solution x = pi/6 + 2pin, where n is an integer.
- 📐 The script discusses finding angles for which cosine x equals negative square root of 2/2, identifying angles at 3pi/4 and 5pi/4 in the unit circle.
- 🔄 For the cosine equation, the general solution is x = 3pi/4 + 2pin and x = 5pi/4 + 2pin, where n is any integer, covering all quadrants where cosine is negative.
- 📉 The script solves an equation involving sine and square root of three, restricting solutions to the range 0 to 2pi, resulting in x = 4pi/3 and 5pi/3.
- 🤔 For tangent-related problems, the script provides a method to find angles for which tangent x equals negative square root of 3/3, identifying angles at 5pi/6 and 11pi/6.
- 🔢 The general solution for tangent equations is given as x = 5pi/6 + pin and x = 11pi/6 + pin, where n is any integer, considering the period of tangent as pi.
- 🔄 The script also solves an equation involving sine squared, resulting in sine x being equal to plus or minus square root of 2/2, covering all four quadrants.
- 🧩 The final solution for the sine squared equation is x = pi/4 + pi/2n, where n is any integer, simplifying the expression to represent all possible angles.
- 📝 The script emphasizes the importance of writing solutions in terms of an integer variable n to account for all possible angles, especially for trigonometric functions with periodicity.

###### Q & A

### What is the value of x if sine x equals one-half?

-The values of x are pi/6 and 5pi/6, which correspond to 30 degrees and 150 degrees respectively. These are the angles where sine is equal to one-half in the unit circle.

### Why are both pi/6 and 5pi/6 solutions to the equation sine x = 1/2?

-Both pi/6 and 5pi/6 are solutions because sine is positive in quadrants one and two. Since sine of 30 degrees (pi/6) is 1/2, and sine is also positive at 150 degrees (5pi/6), both angles satisfy the equation.

### How can you find all possible solutions for the equation sine x = 1/2?

-To find all solutions, you can add multiples of 2pi to the initial solutions (pi/6 and 5pi/6). The general solution is x = pi/6 + 2pi*n or x = 5pi/6 + 2pi*n, where n is any integer.

### What are the angles where cosine x equals negative square root of 2 divided by 2?

-The angles are 135 degrees (3pi/4) and 225 degrees (5pi/4). These angles are in quadrants 2 and 3 where cosine is negative, and the reference angle is 45 degrees.

### How do you determine the general solution for the equation cosine x = -√2/2?

-The general solution is x = 3pi/4 + 2pi*n and x = 5pi/4 + 2pi*n, where n is any integer. This accounts for all coterminal angles that satisfy the equation.

### What is the range of solutions for the equation 2sine x + √3 = 0 when restricted to 0 to 2pi?

-The solutions in the range of 0 to 2pi are x = 4pi/3 and x = 5pi/3. These angles are in the specified range and satisfy the equation.

### How do you find the reference angle for the equation 3tangent x + √3 = 0?

-The reference angle is 30 degrees, as tangent of 30 degrees (pi/6) is equal to √3/3, which is the value we get after solving the equation for tangent x.

### What are the general solutions for the equation 3tangent x + √3 = 0?

-The general solutions are x = 5pi/6 + pi*n and x = 11pi/6 + pi*n, where n is any integer. This accounts for the periodicity of the tangent function, which is pi.

### What does the equation 2sine squared x - 1 = 0 represent?

-The equation represents a situation where the square of the sine function is equal to 1/2. After simplification, it leads to sine x being equal to ±√2/2.

### What are the angles where sine x equals ±√2/2?

-The angles are 45 degrees (pi/4), 135 degrees (3pi/4), 225 degrees (5pi/4), and 315 degrees (7pi/4). These angles are in the quadrants where sine is positive or negative as required.

### How can you express all solutions for the equation 2sine squared x - 1 = 0?

-The general expression for all solutions is x = pi/4 + pi/2*n, where n is any integer. This accounts for the 90-degree difference between the angles where sine x equals ±√2/2.

###### Outlines

##### 📚 Solving Trigonometric Equations with Unit Circle

This paragraph discusses solving trigonometric equations using the unit circle. It begins with the equation sine x = 1/2, identifying angles such as 30° (π/6) and 150° (5π/6) where sine equals 1/2. The concept of coterminal angles is introduced, explaining how adding multiples of 2π to these angles yields all possible solutions for x. The general solution is given as x = π/6 + 2πn and x = 5π/6 + 2πn, where n is any integer. The paragraph then moves on to find solutions for cosine x = -√2/2, identifying angles in quadrants 2 and 3 where cosine is negative. Reference angles of 45° (π/4) and 135° (3π/4) are used to find all solutions, expressed as x = 3π/4 + 2πn and x = 5π/4 + 2πn. The section concludes with an example of solving 2sin(x) + √3 = 0 within the range of 0 to 2π, leading to solutions of x = 4π/3 and x = 5π/3.

##### 🔍 Advanced Trigonometric Equations and Their Solutions

The second paragraph delves into more complex trigonometric equations. It starts with the equation 3tan(x) + √3 = 0, aiming to find all solutions by isolating tan(x) and simplifying to tan(x) = -√3/3. Using the 30-60-90 triangle, the reference angle of 30° is identified, and the angles in quadrants 2 and 4 where tangent is negative are calculated as 150° (5π/6) and 330° (11π/6). The general solution for tangent equations is given as x = 5π/6 + πn and x = 11π/6 + πn, where n is any integer. The paragraph then tackles the equation 2sin^2(x) - 1 = 0, which simplifies to sin^2(x) = 1/2. By taking the square root of both sides, the solutions for sin(x) are ±√2/2, leading to angles in all four quadrants: 45° (π/4), 135° (3π/4), 225° (5π/4), and 315° (7π/4). The final expression for all solutions is x = π/4 + π/2n, where n is any integer, providing a comprehensive method to find all possible values of x.

##### 📐 Understanding Trigonometric Identities and Their Applications

The final paragraph focuses on the trigonometric identity involving sine squared. Starting with the equation 2sin^2(x) - 1 = 0, the paragraph simplifies it to sin^2(x) = 1/2, and then takes the square root of both sides to find sin(x) = ±√2/2. The solutions are analyzed in all four quadrants, where sine is positive in quadrants 1 and 2, and negative in quadrants 3 and 4, leading to the angles 45° (π/4), 135° (3π/4), 225° (5π/4), and 315° (7π/4). To generalize the solution for all possible values of x, the paragraph suggests adding 2πn to each solution, recognizing that each solution differs by π/2. The final expression for all solutions is x = π/4 + π/2n, where n is any integer, offering a simplified way to express all solutions to the trigonometric equation.

###### Mindmap

###### Keywords

##### 💡sine

##### 💡unit circle

##### 💡quadrants

##### 💡coterminal angles

##### 💡cosine

##### 💡reference angle

##### 💡tangent

##### 💡period

##### 💡rationalize

##### 💡solutions

###### Highlights

Sine x equals one-half can be solved using the unit circle, where sine of 30 degrees or pi/6 is equal to one-half.

Sine is positive in quadrants one and two, so x can also be 5 pi/6.

All solutions for sine x = 1/2 can be found by adding 2 pi to pi/6 and 5 pi/6, resulting in 13 pi/6 and 17 pi/6 respectively.

The general equation for all solutions of sine x = 1/2 is x = pi/6 + 2 pi n, where n is any integer.

Cosine x equals negative square root of 2/2 can be solved by considering cosine is negative in quadrants 2 and 3.

Reference angle for cosine 45 is the square root of 2/2, leading to solutions of 135 degrees or 3 pi/4 and 225 degrees or 5 pi/4.

The general solution for cosine x = -sqrt(2)/2 is x = 3 pi/4 + 2 pi n and x = 5 pi/4 + 2 pi n, where n is any integer.

For the equation 2 sine x + sqrt(3) = 0, solutions are restricted from 0 to 2 pi.

Isolating sine x gives sine x = -sqrt(3)/2, and the reference angle is 60 degrees or pi/3.

Solutions to the equation are x = 4 pi/3 and x = 5 pi/3, which are within the specified range.

The equation 3 tangent x + sqrt(3) = 0 leads to tangent x = -sqrt(3)/3.

The reference angle for tangent x = -sqrt(3)/3 is 30 degrees or pi/6, with tangent being negative in quadrants two and four.

Solutions include x = 5 pi/6 and x = 11 pi/6, with the period of tangent being pi.

The general form for solutions is x = 5 pi/6 + pi n and x = 11 pi/6 + pi n, where n is any integer.

For the equation 2 sine squared x - 1 = 0, the solution involves taking the square root of both sides, resulting in sine x = ±sqrt(2)/2.

Sine x can be positive or negative, leading to solutions in all four quadrants: pi/4, 3 pi/4, 5 pi/4, and 7 pi/4.

A general expression for all solutions is x = pi/4 + pi/2 n, where n is any integer, simplifying the solution set.

Alternatively, solutions can be written as adding 2 pi n to each individual solution for mathematical correctness.

###### Transcripts

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