3.3 Using inverse trigonometric functions to solve trigonometric equations

• What do inverse trigonometric functions take as input?

  A ratio
• What is the domain of the inverse sine function, $\sin^{ - 1}(x)$?


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


  $0 < θ < \pi$
• An example of a regular trigonometric function is $sin(30°)$, which equals 0.5
• Match each inverse trigonometric function with its domain:

  $\tan^{ - 1}(x)$ ↔️ Domain: -∞ < x < ∞

  $\cos^{ - 1}(x)$ ↔️ Domain: -1 ≤ x ≤ 1

  $\sec^{ - 1}(x)$ ↔️ 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:

  $\sin^{ - 1}(x)$ ↔️ Domain: -1 ≤ x ≤ 1, Range: $- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}$

  $\cos^{ - 1}(x)$ ↔️ Domain: -1 ≤ x ≤ 1, Range: $0 ≤ θ ≤ \pi$

  $\tan^{ - 1}(x)$ ↔️ Domain: -∞ < x < ∞, Range: $- \frac{\pi}{2} < θ < \frac{\pi}{2}$

  $\cot^{ - 1}(x)$ ↔️ Domain: -∞ < x < ∞, Range: $0 < θ < \pi$
• The domain of $\tan^{ - 1}(x)$ is all real numbers.


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


  $- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}$
• 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, $\cos^{ - 1}(x)$?


  -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