Trigonometry

Fundamental Concepts of Trigonometry

  • Sine, Cosine and Tangent: Basic trigonometric functions, denoted as sin, cos and tan respectively. These are ratios defined in terms of the angles and sides of a right triangle.
  • Cosec, Sec, and Cot: Reciprocal trigonometric functions, defined as cosec=1/sin, sec=1/cos, cot=1/tan.
  • Trigonometric Identities: Important mathematical relationships between trigonometric functions, including Pythagorean, double angle, and half angle identities.
  • Radians and Degrees: Two different systems for measuring angles.
  • Unit Circle: A circle with radius 1, usually used as a tool for linking trigonometry and geometry. The coordinates of a point on the unit circle provide the values of the sine, the cosine, and the tangent of the angle.

Trigonometric Equations and Identities

  • Trigonometric equations: Equations involving trigonometric functions. They can be solved using trigonometric identities, inverse trigonometric functions and other algebraic methods.
  • Pythagorean identities: There are three identities that are derived from the Pythagorean Theorem. These include sin²(θ) + cos²(θ) = 1.
  • Compound angle identities: These identities express the sine, cosine, and tangent of the sum or difference of two angles in terms of sines, cosines, and tangents of the angles.
  • Double and Half Angle identities: These identities express the sine, cosine and tangent of double or half an angle in terms of sines, cosines and tangents of the original angle.

Graphs of Trigonometric Functions

  • Sine and Cosine Graphs: Understand the waveform, period, amplitude and phase shift of these graphs.
  • Tangent Graph: Unlike sine and cosine graphs, tangent graph isn’t periodic, it has vertical asymptotes and no maximum or minimum.
  • Transformations of Trigonometric Graphs: Able to apply transformations including translation, dilation, and reflection to the graphs of trigonometric functions.

Inverse Trigonometric Functions

  • Inverse Sine, Cosine and Tangent: Defined as arcsin, arccos and arctan respectively, or sometimes as sin⁻¹ , cos⁻¹ and tan⁻¹.
  • Properties of Inverse Trigonometric Functions: Understanding the range and domain of these functions.
  • Solving Equations Using Inverse Trigonometric Functions: These functions can be utilized to solve equations and obtain specific angle measures.

Trigonometrical Applications

  • Solving Right-Angled Triangles: The use of sine, cosine and tangent to find unknown angles and lengths.
  • The Sine and Cosine Rules: The rules for any triangle (not just right-angled ones). The sine rule is a/sin(A) = b/sin(B) = c/sin(C) and the cosine rule is c² = a² + b² - 2ab cos(C).
  • Area of Triangle: The area can be found using 1/2absin(C).
  • Simple Harmonic Motion (SHM): Modelling of SHM using sine or cosine functions.