Trigonometric ratios: sin, cos, and tan

Learning Objectives Define sine, cosine, and tangent in the context of a right-angled triangle. Recall and apply the SOH CAH TOA mnemonic to solve for unknown sides and angles. Determine the trigonometric ratios for an angle in standard position given a point on its terminal arm. Calculate the exact values of sin, cos, and tan for special angles (30°, 45°, 60°). Identify the sign (+/-) of sine, cosine, and tangent in each of the four quadrants. Use the Pythagorean identity and quotient identity to find one ratio given another. How does your phone's GPS know your exact location using satellites, or how do game developers create realistic 3D worlds? 🌍 It all comes down to the power of triangles and angles! This tutorial introduces the three fundamental trigonometric rat...

TermDefinitionExample Right-Angled TriangleA triangle with one angle measuring exactly 90°. The side opposite the right angle is the hypotenuse, the longest side. The other two sides are the opposite and adjacent sides, relative to a chosen acute angle (θ).In a triangle with vertices at (0,0), (4,0), and (4,3), the angle at (4,0) is 90°. The hypotenuse is the side connecting (0,0) and (4,3). Relative to the angle at (0,0), the side of length 3 is 'opposite' and the side of length 4 is 'adjacent'. Sine (sin)The ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. In the unit circle, it represents the y-coordinate of a point on the circle.If the opposite side is 3 and the hypotenuse is 5, then sin(θ) = 3/5. Cosine...

SOH CAH TOA \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad | \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad | \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} A mnemonic for remembering the definitions of the three basic trigonometric ratios in a right-angled triangle. It is the foundation for solving problems involving right triangles. Ratios on the Cartesian Plane \sin(\theta) = \frac{y}{r} \quad | \quad \cos(\theta) = \frac{x}{r} \quad | \quad \tan(\theta) = \frac{y}{x} For an angle θ in standard position whose terminal arm passes through the point (x, y), these formulas define the trigonometric ratios. Here, r is the distance from the origin to the point, calculated as r = \sqrt{x^2 + y^2}. Pythagorean Identity \sin...