Trigonometric ratios: find an angle measure

Learning Objectives Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find an unknown angle in a right-angled triangle. Determine the principal value of an angle given a trigonometric ratio. Find all possible angles within a given domain (e.g., 0° to 360°) that satisfy a trigonometric equation. Apply the concept of reference angles to find solutions in all four quadrants. Use a calculator correctly to find angle measures in both degrees and radians. Solve real-world problems that require finding an angle measure using trigonometric ratios. Ever wondered how engineers determine the perfect angle for a wheelchair ramp or how astronomers calculate the position of stars? 📐 It all comes down to finding the missing angle! In previous lessons, you used an angle to find...

TermDefinitionExample Inverse Trigonometric FunctionsFunctions that 'undo' the standard trigonometric functions. They take a ratio as input and return an angle as output. They are denoted as sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x), or alternatively as arcsin(x), arccos(x), and arctan(x).If sin(30°) = 0.5, then sin⁻¹(0.5) = 30°. Principal ValueThe unique angle value returned by an inverse trigonometric function, which lies within a restricted, standardized range. This ensures there is only one output for any given input.The principal value for sin⁻¹(0.5) is 30°, even though sin(150°) is also 0.5. The range for sin⁻¹ is [-90°, 90°]. Reference Angle (α)The acute angle (< 90°) formed by the terminal arm of an angle (θ) in standard position and the horizontal x-axis. It is always positiv...

Inverse Sine (Arcsine) If sin(θ) = x, then θ = sin⁻¹(x) Use this when you know the ratio of the opposite side to the hypotenuse and need to find the angle. The principal value range is [-90°, 90°] or [-π/2, π/2]. Inverse Cosine (Arccosine) If cos(θ) = x, then θ = cos⁻¹(x) Use this when you know the ratio of the adjacent side to the hypotenuse and need to find the angle. The principal value range is [0°, 180°] or [0, π]. Inverse Tangent (Arctangent) If tan(θ) = x, then θ = tan⁻¹(x) Use this when you know the ratio of the opposite side to the adjacent side and need to find the angle. The principal value range is (-90°, 90°) or (-π/2, π/2). Finding All Solutions in 0° ≤ θ < 360° Q1: θ = α Q2: θ = 180° - α Q3: θ = 180° + α Q4: θ = 360° - α After finding the re...