Inverses of trigonometric functions

Learning Objectives Define the inverse trigonometric functions: arcsine, arccosine, and arctangent. Correctly use the notation sin⁻¹, cos⁻¹, and tan⁻¹. Use a scientific calculator to find the measure of an angle given its trigonometric ratio. Solve for an unknown angle in a right-angled triangle using inverse trigonometric functions. Distinguish between an inverse trigonometric function (e.g., sin⁻¹(x)) and a reciprocal function (e.g., 1/sin(x)). Apply inverse trigonometric functions to solve real-world problems involving angles of elevation and depression. You know how to find the length of a ramp, but how do you find out how steep it is? 🤔 Inverse trig functions give us the answer by helping us find the angle! So far, you've used SOH CAH TOA to find the length of a...

TermDefinitionExample Inverse FunctionA function that 'undoes' the action of another function. For example, if a function takes you from A to B, its inverse takes you from B back to A.If f(x) = x + 5, its inverse is f⁻¹(x) = x - 5. If you input 3 into f(x) you get 8. If you input 8 into f⁻¹(x), you get back to 3. Inverse Sine (Arcsine)The function that 'undoes' sine. If you know the ratio of the opposite side to the hypotenuse, inverse sine tells you the angle.If sin(θ) = 0.5, then θ = sin⁻¹(0.5), which means θ = 30°. Inverse Cosine (Arccosine)The function that 'undoes' cosine. If you know the ratio of the adjacent side to the hypotenuse, inverse cosine tells you the angle.If cos(θ) = 0.5, then θ = cos⁻¹(0.5), which means θ = 60°. Inverse Tangent (Arctangent)...

Finding an Angle with Sine If sin(θ) = (Opposite / Hypotenuse), then θ = sin⁻¹(Opposite / Hypotenuse) Use this when you know the lengths of the side opposite the angle you're looking for and the hypotenuse. Finding an Angle with Cosine If cos(θ) = (Adjacent / Hypotenuse), then θ = cos⁻¹(Adjacent / Hypotenuse) Use this when you know the lengths of the side adjacent to the angle you're looking for and the hypotenuse. Finding an Angle with Tangent If tan(θ) = (Opposite / Adjacent), then θ = tan⁻¹(Opposite / Adjacent) Use this when you know the lengths of the side opposite and the side adjacent to the angle you're looking for.