Find trigonometric ratios using the unit circle

Learning Objectives Define the unit circle and relate the coordinates of a point on its circumference to the cosine and sine of the angle in standard position. Identify the coordinates of points on the unit circle corresponding to special angles (0, π/6, π/4, π/3, π/2) and their multiples in all four quadrants. Calculate the values of all six trigonometric functions (sin, cos, tan, csc, sec, cot) for a given angle using the coordinates on the unit circle. Determine the sign (positive or negative) of any trigonometric function in any of the four quadrants. Evaluate trigonometric functions for quadrantal angles (0, π/2, π, 3π/2, 2π). Find the trigonometric ratios for angles greater than 2π and negative angles by identifying their coterminal angles on the unit circle. How can a...

TermDefinitionExample Unit CircleA circle centered at the origin (0, 0) of the Cartesian plane with a radius of exactly 1. Its equation is x² + y² = 1.The point (√2/2, √2/2) lies on the unit circle because (√2/2)² + (√2/2)² = 2/4 + 2/4 = 1. Angle in Standard PositionAn angle whose vertex is at the origin and whose initial side lies along the positive x-axis. Positive angles are measured counter-clockwise, and negative angles are measured clockwise.An angle of 150° starts at the positive x-axis and rotates counter-clockwise into the second quadrant. Terminal PointThe point P(x, y) where the terminal side of an angle θ in standard position intersects the unit circle.For an angle of π/2 (or 90°), the terminal side lies on the positive y-axis, so the terminal point on the unit circle is (0, 1...

Primary Trigonometric Ratios on the Unit Circle For an angle θ whose terminal side intersects the unit circle at P(x, y): \n cos(θ) = x \n sin(θ) = y \n tan(θ) = y/x, for x ≠ 0 These formulas define the core trigonometric ratios directly from the coordinates of the terminal point. The x-coordinate is the cosine, the y-coordinate is the sine, and their ratio is the tangent. Reciprocal Trigonometric Ratios on the Unit Circle For the same terminal point P(x, y): \n sec(θ) = 1/x, for x ≠ 0 \n csc(θ) = 1/y, for y ≠ 0 \n cot(θ) = x/y, for y ≠ 0 These are the reciprocal identities. Secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent. They are found by taking the reciprocal of the coordinates or their ratio. Quadrant...