Podcast
Questions and Answers
What is the primary reason for converting all terms in a trigonometric equation to one side before solving?
What is the primary reason for converting all terms in a trigonometric equation to one side before solving?
- To simplify the equation by reducing the number of terms.
- To ensure all terms have the same degree.
- To make the equation easier to graph.
- To avoid the common mistake of incorrectly canceling terms, which can lead to loss of valid solutions. (correct)
If $\cos(x) = -\frac{\sqrt{3}}{2}$, and $x$ is between $0$ and $2\pi$, in which quadrants could $x$ lie?
If $\cos(x) = -\frac{\sqrt{3}}{2}$, and $x$ is between $0$ and $2\pi$, in which quadrants could $x$ lie?
- Quadrants III and IV
- Quadrants II and III (correct)
- Quadrants I and II
- Quadrants I and IV
In solving $\sin^2(x) = \sin(x)$, why is it important to avoid dividing both sides by $\sin(x)$?
In solving $\sin^2(x) = \sin(x)$, why is it important to avoid dividing both sides by $\sin(x)$?
- Dividing by $\sin(x)$ would make the equation undefined.
- Dividing by $\sin(x)$ could eliminate solutions where $\sin(x) = 0$. (correct)
- Dividing by $\sin(x)$ could introduce extraneous solutions.
- Dividing by $\sin(x)$ would change the period of the function.
When solving a trigonometric equation that includes both $\sin(x)$ and $\cos^2(x)$, what is a common strategy to simplify the equation?
When solving a trigonometric equation that includes both $\sin(x)$ and $\cos^2(x)$, what is a common strategy to simplify the equation?
If you find that $\sin(x) = 1.5$ when solving a trigonometric equation, what should you conclude?
If you find that $\sin(x) = 1.5$ when solving a trigonometric equation, what should you conclude?
What is the general strategy for solving trigonometric equations that are quadratic in form?
What is the general strategy for solving trigonometric equations that are quadratic in form?
If $\cos(x) = \frac{1}{2}$ and $\sin(x) = -\frac{\sqrt{3}}{2}$, what is the value of $x$ in the interval $[0, 2\pi)$?
If $\cos(x) = \frac{1}{2}$ and $\sin(x) = -\frac{\sqrt{3}}{2}$, what is the value of $x$ in the interval $[0, 2\pi)$?
What is the solution set for the equation $2\cos(x) - 1 = 0$ in the interval $[0, 2\pi)$?
What is the solution set for the equation $2\cos(x) - 1 = 0$ in the interval $[0, 2\pi)$?
Flashcards
Solving Trigonometric Equations
Solving Trigonometric Equations
Finding the angle(s) that satisfy a trigonometric equation.
Solve sin(x) = √2/2
Solve sin(x) = √2/2
sin(x) = √2/2 has solutions at π/4 and 3π/4 within 0 to 2π.
Solve cos²(x) = 1/4
Solve cos²(x) = 1/4
cos²(x) = 1/4 has solutions at π/3, 2π/3, 4π/3, and 5π/3 within range 0 to 2π.
Factoring in Trig Equations
Factoring in Trig Equations
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Solve sin²(x) = sin(x)
Solve sin²(x) = sin(x)
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Using Trig Identities
Using Trig Identities
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Solve cos²(x) - 2sin(x) + 2 = 0
Solve cos²(x) - 2sin(x) + 2 = 0
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Range of Sine and Cosine
Range of Sine and Cosine
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Study Notes
Solving Trigonometric Equations
- Trigonometric equations can be solved
- Sharing the content is encouraged to support channel growth
Equation 1: sin(x) = √2/2
- Solve sin(x) = √2/2 to find values of x in radians
- The values should be between 0 and 2π
- x = 45 degrees or π/4 radians satisfies the equation
- Sine is positive in the first and second quadrants on the unit circle
- √2/2 corresponds to 45 degrees in the first quadrant
- Reflecting the angle from the first quadrant finds the second quadrant solution
- The second quadrant solution is 180 - 45 = 135 degrees (3π/4 radians)
- Solutions: x = π/4 or x = 3π/4
Equation 2: cos²(x) = 1/4
- The equation is cos²(x) = 1/4
- Taking the square root of both sides results in cos(x) = ±√(1/4) = ±1/2
- Positive and negative values must be considered
- Squaring either value results in a positive number and therefore a valid solution
- Find angles where the cosine equals +1/2 and -1/2
- Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants
- cos(x) = 1/2 corresponds to 60 degrees in the first quadrant
- Angles such as 120, 240 and 300 can be found
- Solutions: 60 degrees (π/3), 120 degrees (2π/3), 240 degrees (4π/3), and 300 degrees (5π/3)
Common Mistake: Canceling Terms
- Avoid dividing or canceling terms
- Instead, rearrange to one side and factor
Equation 3: sin²(x) = sin(x)
- The equation is sin²(x) = sin(x)
- Move all terms to one side: sin²(x) - sin(x) = 0
- Factor out sin(x): sin(x) * (sin(x) - 1) = 0
- Set each factor to zero: sin(x) = 0 or sin(x) = 1
- Solutions for sin(x) = 0 are x = 0, 180 degrees, and 360 degrees
- Solution for sin(x) = 1 is x = 90 degrees
Equation 4: cos²(x) - 2sin(x) + 2 = 0
- The equation is cos²(x) - 2sin(x) + 2 = 0
- Use the identity cos²(x) = 1 - sin²(x) to convert to a single trigonometric function
- Substitute: 1 - sin²(x) - 2sin(x) + 2 = 0
- Simplify: -sin²(x) - 2sin(x) + 3 = 0
- The quadratic formula can find the roots of sin(x)
- Two solutions are found sin(x) = -3 and sin(x) =1, however sine must be between -1 and 1
- Discard sin(x) = -3 as it's outside the possible range [-1, 1]
- Solution is sin(x) = 1, which occurs at x = 90 degrees or π/2 radians
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