Trigonometry
Free lessons and practice worksheets for trigonometry — worked examples, common mistakes, and step-by-step solutions.
- 01Trigonometry3 min read
3D Trigonometry
3D trigonometry applies trigonometric principles to three-dimensional objects, calculating distances, angles, and spatial relationships that cannot be solved using 2D methods alone. The most fundamental calculation involves finding the space diagonal of a rectangular box using the formula d = √(l² + w² + h²), which extends the Pythagorean theorem into three dimensions. Problems typically involve rectangular prisms, pyramids, and other geometric solids where measurements span across multiple planes.
- 02Trigonometry3 min read
Inverse Trigonometry
Inverse trigonometric functions reverse the action of sine, cosine, and tangent by finding the angle that produces a given ratio. The three primary inverse functions are arcsin, arccos, and arctan, each with specific output ranges called principal values. These functions appear in CCSS.HSF.TF.B.6 as tools for solving trigonometric equations and analyzing periodic phenomena.
- 03Trigonometry3 min read
Sine & Cosine Rules
The sine and cosine rules are fundamental trigonometric formulas used to solve triangles when not all sides and angles are known. The sine rule states that a/sin(A) = b/sin(B) = c/sin(C), connecting each side to the sine of its opposite angle. The cosine rule, c² = a² + b² - 2ab·cos(C), provides a way to find unknown sides or angles using the Pythagorean theorem with an additional correction term.
- 04Trigonometry3 min read
Trigonometric Equations
Trigonometric equations are mathematical equations that contain trigonometric functions like sine, cosine, or tangent and require finding all angle values that satisfy the equation. These equations typically have multiple solutions within a given interval due to the periodic nature of trigonometric functions. Standard practice involves solving over intervals like [0°, 360°] or [0, 2π] radians.
- 05Trigonometry3 min read
Trigonometric Graphs
Trigonometric graphs represent periodic functions that repeat their values in regular intervals, with sine and cosine functions producing characteristic wave patterns. The general form y = A sin(Bx + C) + D contains four parameters that control the wave's amplitude, period, horizontal shift, and vertical position. These graphs model cyclical phenomena ranging from sound waves to seasonal temperature variations.
- 06Trigonometry3 min read
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that remain true for all values of the variable where both sides are defined. The most fundamental identity is sin²x + cos²x = 1, known as the Pythagorean identity. These relationships form the foundation for simplifying complex trigonometric expressions and solving equations across mathematics, physics, and engineering.
- 07Trigonometry3 min read
Trigonometry (SOH CAH TOA)
Trigonometry uses ratios to relate the angles and sides of right triangles through three fundamental functions: sine, cosine, and tangent. The mnemonic SOH CAH TOA helps recall these relationships: SOH means sine equals opposite over hypotenuse, CAH means cosine equals adjacent over hypotenuse, and TOA means tangent equals opposite over adjacent. These ratios form the foundation for solving problems involving right triangles and appear in CCSS.HSG.SRT standards for geometric problem-solving.
- 08Trigonometry3 min read
Unit Circle
The unit circle provides exact values for trigonometric functions at standard angles, eliminating the need for decimal approximations. On a circle with radius 1 centered at the origin, the coordinates (x, y) of any point correspond to (cos θ, sin θ) for angle θ. Standard angles like 30°, 45°, 60°, and their multiples produce exact values involving square roots and simple fractions.