12.1 Solving Trigonometric Equations

Solving trigonometric equations is like cracking a code. You'll use algebraic tricks and inverse functions to unlock the secrets of sine, cosine, and tangent. Remember, these functions repeat, so you might find multiple solutions hiding in different intervals.

Graphing can be your secret weapon. By plotting both sides of the equation, you can spot where they cross and confirm your answers. Don't forget to use trigonometric identities – they're like shortcuts that can simplify tricky equations and make solving a breeze.

Solving trigonometric equations

Algebraic techniques and inverse trigonometric functions

Top images from around the web for Algebraic techniques and inverse trigonometric functions

• 

  Inverse Trigonometric Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Trigonometric Functions ‹ OpenCurriculum View original

  Is this image relevant?

• 

  Inverse Trigonometric Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Functions | Algebra and Trigonometry View original

  Is this image relevant?

1 of 3

Top images from around the web for Algebraic techniques and inverse trigonometric functions

• 

  Inverse Trigonometric Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Trigonometric Functions ‹ OpenCurriculum View original

  Is this image relevant?

• 

  Inverse Trigonometric Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Inverse Functions | Algebra and Trigonometry View original

  Is this image relevant?

1 of 3

• Isolate the trigonometric function on one side of the equation and apply the inverse trigonometric function to both sides
• Consider the domain and range of the trigonometric functions involved when solving equations
• Use algebraic techniques such as factoring, simplifying, and substitution to manipulate trigonometric equations before applying inverse trigonometric functions
• Adjust solutions obtained using inverse trigonometric functions by adding or subtracting multiples of π or 2π to find all possible solutions within a given interval (0 ≤ θ ≤ 2π)

Solutions in intervals

Periodicity and number of solutions

• The number of solutions for a trigonometric equation in a given interval depends on the periodicity of the trigonometric functions involved
  • Sine and cosine functions have a period of 2π
  • Tangent and cotangent functions have a period of π
• Consider the period of the trigonometric function and the length of the interval to determine the number of solutions
  • If the interval is equal to or greater than the period, there will be at least one solution per period within the interval
  • If the interval is smaller than the period, there may be zero, one, or more solutions, depending on the specific equation and the values of the trigonometric function within the interval

Solving equations in specific intervals

• When solving trigonometric equations in a specific interval, consider the following steps:
  1. Solve the equation using algebraic techniques and inverse trigonometric functions
  2. Identify the period of the trigonometric function involved
  3. Determine the number of solutions based on the period and the length of the interval
  4. Adjust the solutions obtained in step 1 by adding or subtracting multiples of the period to ensure they fall within the given interval
• Example: Solve sin(θ) = 0.5 in the interval [0, π]
  1. θ = arcsin(0.5) ≈ 0.524 or θ = π - arcsin(0.5) ≈ 2.618
  2. The period of the sine function is 2π
  3. The interval [0, π] is half the period, so there will be one solution
  4. The solution θ ≈ 0.524 falls within the interval, while θ ≈ 2.618 does not, so the only solution is θ ≈ 0.524

Graphical verification of solutions

Intersection points as solutions

• Graph the left-hand side (LHS) and right-hand side (RHS) of a trigonometric equation to verify the solutions obtained algebraically
• The points of intersection between the LHS and RHS graphs represent the solutions to the trigonometric equation
• Consider the domain and range of the trigonometric functions and any restrictions imposed by the equation when verifying solutions graphically

Identifying extraneous solutions

• Graphing can help identify extraneous solutions that may arise from algebraic manipulations or the use of inverse trigonometric functions
• Extraneous solutions are values that satisfy the manipulated equation but do not satisfy the original equation
• Example: Solve cos(θ) = -2
  • Algebraically, cos(θ) = -2 has no solution, as the cosine function's range is [-1, 1]
  • If the equation is manipulated to θ = arccos(-2), graphing will show that there are no points of intersection between y = cos(θ) and y = -2, confirming that there are no solutions

Trigonometric identities for solving equations

Simplifying and solving equations using identities

• Apply trigonometric identities, such as the Pythagorean identity, double-angle formulas, and sum-to-product formulas, to simplify and solve trigonometric equations
• Applying trigonometric identities can help reduce the complexity of the equation, making it easier to solve using algebraic techniques or inverse trigonometric functions
• Maintain the equality of the equation and ensure that the domain of the functions remains consistent when using trigonometric identities

Common trigonometric identities

• Pythagorean identity: $sin^2(θ) + cos^2(θ) = 1$
• Double-angle formulas: $sin(2θ) = 2sin(θ)cos(θ)$, $cos(2θ) = cos^2(θ) - sin^2(θ)$
• Sum-to-product formulas: $sin(α) + sin(β) = 2sin((α+β)/2)cos((α-β)/2)$
• Applying trigonometric identities may introduce additional solutions or change the form of the equation, so verify the solutions and ensure they satisfy the original equation
• Example: Solve $sin^2(θ) - sin(θ) = 0$ using the Pythagorean identity
  1. Substitute $cos^2(θ) = 1 - sin^2(θ)$ into the equation: $(1 - cos^2(θ)) - sin(θ) = 0$
  2. Simplify: $1 - cos^2(θ) - sin(θ) = 0$
  3. Factor: $(1 - sin(θ))(1 + sin(θ)) - cos^2(θ) = 0$
  4. Solve for $sin(θ)$: $sin(θ) = 0$ or $sin(θ) = 1$
  5. Solve for $θ$: $θ = 0$ or $θ = π$ (for $sin(θ) = 0$), and $θ = π/2$ (for $sin(θ) = 1$)