Trigonometric Functions and Their Inverse Functions Study Notes

Trigonometric Functions and Their Inverse Functions

Understanding the Cosine Function

  • Problem: Given that $ ext{cos}(x) = rac{1}{3}$, find the values of $x$.
    • Quadrant Analysis:
    • Cosine is positive in the first quadrant (A) and the fourth quadrant (C).
    • Solutions in Quadrants:
    • First solution will be in the first quadrant.
    • Second solution in the fourth quadrant.
    • Calculating the Angles:
    • To find the first angle, compute $ ext{cos}^{-1}igg( rac{1}{3}igg)$, which is approximately $70.5^ ext{°}$.
    • For the second solution, subtract from 360 degrees:
      • $360^ ext{°} - 70.5^ ext{°} = 289.5^ ext{°}$.

General Properties of Cosine

  • If $ ext{cos}(x) = k$ where $k$ is positive, the corresponding angles will appear in both the first and fourth quadrants.
  • For $ ext{cos}(x) = -k$, the angles will be found in the second and third quadrants.

Understanding the Sine Function

  • Example: Given $ ext{sin}(x) = rac{1}{2}$,
    • Identify quadrants where sine is positive: first (A) and second (S).
    • The value of $ ext{sin}^{-1}igg( rac{1}{2}igg)$ yields $30^ ext{°}$.
    • Possible angles and their calculations:
    • First Quadrant: $30^ ext{°}$.
    • Second Quadrant: $180^ ext{°}-30^ ext{°} = 150^ ext{°}$.

Working with the Tangent Function

  • Given: $ ext{tan}(x) = -1$,
    • Tangent is negative in the second (S) and fourth (C) quadrants.
  • Solution Steps:
    • Drop the negative sign and find $ ext{tan}^{-1}(1)$, which equals $45^ ext{°}$.
    • Translating this into quadrant-specific conclusions:
    • In the second quadrant: $180^ ext{°}-45^ ext{°} = 135^ ext{°}$.
    • In the fourth quadrant: $360^ ext{°}-45^ ext{°} = 315^ ext{°}$.

The All Students Take Calculus Rule

  • A mnemonic to remember the signs of trigonometric functions in quadrants:
    • A = All (All positive), S = Sine (positive), T = Tangent (positive), C = Cosine (positive).

Impact of Quadrants on Solutions

  • For Negative Values:
    • Always identify the quadrants where the sign is negative to find possible angles.
  • Secondary Solutions: For angles found, always calculate their supplementary or complementary counterparts based on the quadrant rules.

Working with Defined Intervals

  • Determine if given angles must fit within defined intervals (e.g., between 0 and 180 degrees).
  • Example: For cosine results in the interval $[0, 180]$, if cosine yields two potential angles, confirm which are valid considering the interval restrictions.
    • If required to report angles in $[0, 180]$, any calculated angle beyond 180 must be excluded from the final answer set.

Doubling the Angle

  • Working from $ ext{sin}(2x)$ means greater scrutiny over intervals since adjustments must be made during computations:
    • Adjust input intervals accordingly; for example, doubling the range to identify answers for $2x$ and dividing them post-calculation to yield $x$ results, maintaining computational reliability.

Examples for Further Practice

  • Solve problems such as:
    • $ ext{cos}(x) = rac{1}{2}$ in the interval $[0, 720]$
    • $ ext{sin}(x) = - rac{1}{3}$ in a specified interval, noting how many solutions might exist depending on positive/negative identification in quadrants.

Conclusion

  • The ability to correctly navigate trigonometric identities relies heavily on quadrant understanding and the strategic application of inverse functions. The technique of adding angles like 360 degrees for subsequent cycles is crucial for establishing all potential solutions.

Final Notes

  • Each trigonometric function has specific behavior according to its angle base: keep the quadrantal impacts in mind for predictions of sign and angle validity.
  • Understanding and applied knowledge are foundational: Transform calculations quickly into visual quadrant analysis helps reinforce understanding and application.