Sin, cos, and tan of special angles

Learning Objectives Recall and state the exact sine, cosine, and tangent values for 30°, 45°, and 60° (and their radian equivalents π/6, π/4, π/3). Derive the trigonometric ratios of special angles using the 45-45-90 and 30-60-90 special right triangles. Identify the coordinates on the unit circle corresponding to special angles in all four quadrants. Calculate the exact values of trigonometric expressions involving special angles, including quadrantal angles (0°, 90°, 180°, 270°). Apply the concept of reference angles and the ASTC rule to find trigonometric values for angles greater than 90° or less than 0°. Solve simple trigonometric equations where the solution is a special angle. Ever wondered how video game developers create realistic movements or how engineers design p...

TermDefinitionExample Special AnglesA set of common angles (0°, 30°, 45°, 60°, 90° and their multiples) that have simple, exact trigonometric ratios which can be expressed with integers and square roots.The angle 60° is a special angle. Its cosine value is exactly 1/2, not an approximation like 0.5. Unit CircleA circle with a radius of 1 centered at the origin of the Cartesian plane. For any angle θ, the point (x, y) where the terminal side of the angle intersects the circle gives the values (cos θ, sin θ).At 90°, the terminal side intersects the unit circle at the point (0, 1). Therefore, cos(90°) = 0 and sin(90°) = 1. Reference Angle (θ')The acute angle (< 90°) formed by the terminal side of an angle θ and the horizontal x-axis. It is always positive.The reference angle for 150°...

45-45-90 Triangle Ratios sin(45°) = \frac{\sqrt{2}}{2}, cos(45°) = \frac{\sqrt{2}}{2}, tan(45°) = 1 These values are derived from an isosceles right triangle whose sides are in the ratio 1 : 1 : √2. Use these for any angle with a reference angle of 45° or π/4. 30-60-90 Triangle Ratios sin(30°) = \frac{1}{2}, cos(30°) = \frac{\sqrt{3}}{2}, tan(30°) = \frac{\sqrt{3}}{3} \| sin(60°) = \frac{\sqrt{3}}{2}, cos(60°) = \frac{1}{2}, tan(60°) = \sqrt{3} Derived from a 30-60-90 triangle with side ratios 1 : √3 : 2. Note that sin(30°) = cos(60°) and cos(30°) = sin(60°). Quadrant Signs (ASTC Rule) Q1: All positive. Q2: Sin positive. Q3: Tan positive. Q4: Cos positive. A mnemonic ('All Students Take Calculus') to remember which trigonometric functions are positive in ea...