Inverses of sin, cos, and tan

Learning Objectives Define the domain and range for the inverse sine, cosine, and tangent functions. Explain why domain restrictions are necessary to create inverse trigonometric functions. Evaluate inverse trigonometric expressions for exact values found on the unit circle. Use a calculator to find approximate values for inverse trigonometric expressions in both radians and degrees. Solve simple trigonometric equations using inverse functions to find a principal value. Evaluate composite functions involving trigonometric and inverse trigonometric functions (e.g., cos(tan⁻¹(x))). Recognize the graphs of y = sin⁻¹(x), y = cos⁻¹(x), and y = tan⁻¹(x). If you know a right triangle has a side opposite an angle that is half the length of the hypotenuse, how can you find the exac...

TermDefinitionExample Inverse FunctionA function that reverses the action of another function. If a function f takes an input x to an output y (f(x) = y), its inverse, denoted f⁻¹(y), takes y back to x (f⁻¹(y) = x).If f(x) = 2x, then its inverse is f⁻¹(x) = x/2. For instance, f(3) = 6 and f⁻¹(6) = 3. Domain RestrictionThe process of limiting the domain (the set of input values) of a function so that it becomes one-to-one, meaning each output corresponds to only one input. This is necessary for a function to have a well-defined inverse.The function y = x² is not one-to-one. By restricting its domain to x ≥ 0, it becomes one-to-one, and its inverse is y = √x. Arcsine (sin⁻¹)The inverse function of sine. It takes a ratio from [-1, 1] and returns the corresponding angle in the restricted rang...

Arcsine Definition & Range y = sin⁻¹(x) <=> sin(y) = x | Range: -π/2 ≤ y ≤ π/2 This defines the relationship between sine and arcsine. The output of arcsine, y, must be an angle in Quadrant I or IV (including the y-axis). Arccosine Definition & Range y = cos⁻¹(x) <=> cos(y) = x | Range: 0 ≤ y ≤ π This defines the relationship between cosine and arccosine. The output of arccosine, y, must be an angle in Quadrant I or II (including the x-axis). Arctangent Definition & Range y = tan⁻¹(x) <=> tan(y) = x | Range: -π/2 < y < π/2 This defines the relationship between tangent and arctangent. The output of arctangent, y, must be an angle in Quadrant I or IV (excluding the endpoints -π/2 and π/2).