Study Notes on Trigonometric Equations and Identities

Understanding Identities and Equations

  • An identity is an equation that holds true for all values of the variable.

    • Any value plugged into an identity will yield a true statement.

    • Example: If an equation was given as an identity, every substitution into the variable makes it valid.

  • An equation, on the other hand, has conditional solutions.

    • Only certain substitutions render the equation true.

Importance of Periodicity in Trigonometric Functions

  • Periodic graphs: The graphs of trigonometric functions repeat values at regular intervals.

    • For example, the sine function has a period of 360 degrees.

    • Within one period, particularly for sine, multiple angles will result in the same output.

  • Example of angles that result in the sine being equal to $ rac{1}{2}$:

    • 30 degrees (in quadrant 1)

    • 150 degrees (in quadrant 2)

Solving Trigonometric Equations

Basic Trigonometric Equation Structure

  • A basic trigonometric equation is where the trigonometric function (e.g., sine, cosine, tangent) is isolated.

  • The goal of solving any equation is to isolate the variable.

    • This specifically means isolating the trigonometric expression (e.g., $ ext{sine}(x)$) or ($ ext{sine}( heta)$).

Example Problem

  • Given $ ext{sine}(x) = rac{1}{2}$, determine the solutions in one period.

    • Sine is positive in quadrants 1 and 2.

    • Find the reference angles for the solutions.

    • Quadrant 1: $30$ degrees

    • Quadrant 2: $180^ ext{o} - 30^ ext{o} = 150^ ext{o}$

  • Both angles represent the solutions to the equation in one period.

Finding All Solutions

  • Trigonometric functions are periodic, meaning they repeat indefinitely.

  • To find all solutions, add multiples of the period to the basic solutions.

    • The period for sine and cosine functions is $360^ ext{o}$.

    • Therefore, to represent all solutions for $ ext{sine}(x) = rac{1}{2}$:

    • 30 + 360k

    • 150 + 360k

    • Here, k is any integer (0, ±1, ±2, …).

Specific Solutions

  • To find specific solutions within a certain interval (e.g., between $0^ ext{o}$ and $360^ ext{o}$):

    • Plug in values for k into the general solutions.

    • For k = 0:

      • Solutions are $30^ ext{o}$ and $150^ ext{o}$.

    • For k = 1:

      • Add $360^ ext{o}$ to each to get $390^ ext{o}$ and $510^ ext{o}$, respectively.

    • May also plug in negative k values for more solutions.

Reference Angles and Quadrants

  • Reference angles are necessary for finding solutions in non-standard quadrants.

  • To find reference angles:

    • Use the positive output value at first for input into $ ext{inverse sine}$ functions.

    • After determining the reference angle, find angles in appropriate quadrants based on the sign of the function.

  • Example: Finding solutions to $ ext{sine}( heta) = - rac{ ext{sqrt}{2}}{2}$

    • Reference angle: $45^ ext{o}$ (as sine is equal to $ rac{ ext{sqrt}{2}}{2}$ in the first quadrant).

    • Final angles: Quadrants 3 and 4 would yield $225^ ext{o}$ and $315^ ext{o}$, respectively.

Special Cases in Trigonometric Functions

Implications of Quadrantal Angles

  • A quadrantal angle occurs at specific intervals generating an exact output.

    • Example: $ ext{cos}( heta) = -1$, which yields:

    • $ heta = 180^ ext{o}$ (no other answer, indicates unique case).

Non-standard Values

  • For values that do not yield common trigonometric angles, like $ ext{cos}( heta) = rac{1}{4}$:

    • Use a calculator to derive approximate angle values.

    • Example: Find $ ext{inverse cos}(1/4)$ which may yield approximately $75.5^ ext{o}$ as a reference angle.

Understanding the Period of Functions

Differences Between Functions

  • Period of $tan( heta)$ is $180^ ext{o}$, thus representation of solutions will differ:

    • General form: 45^ ext{o} + 180^ ext{o}k

  • This equation captures and simplifies all possible periodic solutions across both quadrants effectively.

Solving Complex Equations

  • When dealing with more complex equations involving products:

    • Set each factor to zero (e.g., $a imes b = 0$ implies $a = 0$ or $b = 0$).

  • Example: For the equation $2x^2 - x - 1 = 0$, apply factoring techniques to isolate and solve for x.

Conclusion and Homework

  • Aim to achieve isolated trigonometric equations first and then find all solutions before narrowing down specific solution intervals as needed.

  • Practice problems provided, focus on becoming comfortable with the method of producing both general and specific solutions dynamically.