3.3 Using inverse trigonometric functions to solve trigonometric equations

    Cards (55)

    • What do inverse trigonometric functions take as input?
      A ratio
    • What is the domain of the inverse sine function, sin1(x)\sin^{ - 1}(x)?

      -1 ≤ x ≤ 1
    • What is the range of the inverse cotangent function, cot1(x)\cot^{ - 1}(x)?

      0<θ<π0 < θ < \pi
    • An example of a regular trigonometric function is sin(30°)sin(30°), which equals 0.5
    • Match each inverse trigonometric function with its domain:
      tan1(x)\tan^{ - 1}(x) ↔️ Domain: -∞ < x < ∞
      cos1(x)\cos^{ - 1}(x) ↔️ Domain: -1 ≤ x ≤ 1
      sec1(x)\sec^{ - 1}(x) ↔️ Domain: x ≤ -1 or x ≥ 1
    • What is the range of the inverse tangent function, `tan⁻¹(x)`?
      -π/2 < θ < π/2
    • What is the range of the inverse cotangent function, `cot⁻¹(x)`?
      0 < θ < π
    • Steps to solve trigonometric equations using inverse trigonometric functions
      1️⃣ Isolate trigonometric expressions in equations
      2️⃣ Apply appropriate inverse trigonometric functions
      3️⃣ Check solutions for validity
    • Match each inverse trigonometric function with its domain and range:
      sin1(x)\sin^{ - 1}(x) ↔️ Domain: -1 ≤ x ≤ 1, Range: π2θπ2- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}
      cos1(x)\cos^{ - 1}(x) ↔️ Domain: -1 ≤ x ≤ 1, Range: 0θπ0 ≤ θ ≤ \pi
      tan1(x)\tan^{ - 1}(x) ↔️ Domain: -∞ < x < ∞, Range: π2<θ<π2- \frac{\pi}{2} < θ < \frac{\pi}{2}
      cot1(x)\cot^{ - 1}(x) ↔️ Domain: -∞ < x < ∞, Range: 0<θ<π0 < θ < \pi
    • The domain of tan1(x)\tan^{ - 1}(x) is all real numbers.

      True
    • What is the range of the inverse sine function, sin1(x)\sin^{ - 1}(x)?

      π2θπ2- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}
    • The range of the inverse cosine function, `cos⁻¹(x)`, is 0 ≤ θ ≤ π
    • The range of the inverse secant function, `sec⁻¹(x)`, is 0 ≤ θ ≤ π
    • What is the output of an inverse trigonometric function?
      An angle
    • The domains and ranges of inverse trigonometric functions must be considered when solving trigonometric equations.

      True
    • What is the domain of `sin⁻¹(x)`?
      -1 ≤ x ≤ 1
    • The first step in solving trigonometric equations is to isolate the trigonometric expression
    • In the equation `3sin(x) = 4 - 2`, the next step is to simplify the right side to `3sin(x) = 2
    • To solve for `sin(x) = 0.7`, the appropriate inverse function to apply is sin^-1
    • `sin(x)` is positive in the first and second quadrants.

      True
    • The ranges of inverse trigonometric functions are unlimited.
      False
    • Steps to isolate a trigonometric expression in an equation
      1️⃣ Simplify the equation
      2️⃣ Use algebraic operations to move non-trigonometric terms
      3️⃣ Divide by the coefficient of the trigonometric expression
    • What is the first step in solving a trigonometric equation using inverse functions?
      Isolate the trigonometric expression
    • What do regular trigonometric functions take as input and return as output?
      Angle as input, ratio as output
    • The range of `cos⁻¹(x)` is 0 ≤ θ ≤ π
    • Algebraic operations are used to move non-trigonometric terms in trigonometric equations.

      True
    • The domain and range of inverse trigonometric functions must be considered to ensure solutions are valid.

      True
    • What interval are we considering for solutions in the equation `sin(x) = 0.7`?
      [0, 2π]
    • What is the second solution for `sin(x) = 0.7` in the interval `[0, 2π]`?
      2.366
    • What is the next step after isolating a trigonometric expression in an equation with multiple trigonometric functions?
      Apply the inverse function
    • What type of trigonometric functions are used to solve equations with multiple trigonometric expressions?
      Inverse trigonometric functions
    • What is the isolated trigonometric expression in the equation `2cos(x) - sin(x) = 1`?
      cos(x) = (1 + sin(x))/2
    • Regular trigonometric functions take an angle as input and return a ratio
    • The range of the inverse sine function is -\frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}</latex>
      True
    • What is the domain of the inverse cosine function, cos1(x)\cos^{ - 1}(x)?

      -1 ≤ x ≤ 1
    • What is the domain of the inverse sine function, `sin⁻¹(x)`?
      -1 ≤ x ≤ 1
    • The domain of `csc⁻¹(x)` is `x ≤ -1` or `x ≥ 1`.
      True
    • The domains of inverse trigonometric functions are defined in terms of ratios, not angles.

      True
    • To isolate `sin(x)` in the equation `3sin(x) + 2 = 4`, the final step is to divide both sides by 3
    • Since `sin(x)` is positive in the first and second quadrants, the second solution for `sin(x) = 0.7` is π - 0.775