Trigonometry

Basics of Trigonometry

  • Familiarise yourself with the basic trigonometric ratios: sine, cosine, and tangent.
  • Understand the connection of these ratios to a right-angled triangle, such as how sine (sin) is the ratio of the length of the side opposite an angle to the length of the hypotenuse.
  • Similarily, learn that cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse, and tangent (tan) is the ratio of the length of the side opposite an angle to the length of the side adjacent.
  • Memorise the ratios in terms of sine, cosine, and tangent, using SOHCAHTOA as an effective mnemonic tool.

Trigonometric Identities

  • Learn and apply the Pythagorean identity: sin²x + cos²x = 1.
  • Understand how to derive other identities from this fundamental identity, such as 1 + tan²x = sec²x.
  • Know the reciprocal identities: cosecant (csc), secant (sec), cotangent (cot), which are the reciprocals of sine, cosine, and tangent, respectively.
  • Note the co-function identities for acute angles, such as sin(x) = cos(90 - x).

Trigonometric Functions and their Graphs

  • Recognise the periodic nature of trigonometric functions and be able to sketch the graph of sinx, cosx and tanx.
  • Understand the significance of the amplitude and period in these graphs.
  • Identify phase shifts and transformations in the graphs.

Trigonometric Equations

  • Be equipped to solve simple trigonometric equations by understanding the different values of x that make the equation true within a given range.
  • Be able to use inverse trigonometric operations (arcsin, arccos, arctan) to find specific angles.
  • Apply the Cast rule or the “All-Sine-Tan-Cos” method to determine the sign of a trigonometric function in each quadrant.

Solving Triangles

  • Use the sine rule (a/sinA = b/sinB = c/sinC) to find unknown sides or angles in any triangle.
  • Similarly, apply the cosine rule (c² = a² + b² - 2abcosC) when two sides and the included angle are known or when three sides are known.
  • Know when to use the right triangle trigonometry, sine rule, or cosine rule by properly analyzing the given information in a question.

Real World Trigonometry

  • Utilise knowledge of trigonometry to solve real-life problems involving heights and distances.
  • Apply principles of trigonometry in contexts involving navigation, physics, engineering, and architecture.
  • Combine trigonometric knowledge with other mathematical concepts for complex problem-solving and interpretation.