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MathidiaMathematics Academy
Trigonometry and radians
Lesson 23 of 75
Standard practice

Fundamental identities

Objective. Use tan θ=sin θ/cos θ and sin²θ+cos²θ=1.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Fundamental identities is developed as connected advanced mathematics rather than a memorised recipe. Extend trigonometry to radian measure, identities and complete solution sets. Begin by identifying the objects, definitions, assumptions and domain restrictions that control the problem. Move between symbolic, graphical, numerical and contextual representations, explaining why each transformation preserves the required meaning. Compare efficient methods, retain exact values until approximation is justified, and use notation that makes every dependency visible. The final stage is to test limiting cases, dimensions, signs and alternative representations so the conclusion is both rigorous and useful.

Worked example

Select a point on y=sin x represented schematically by its origin crossing.

y = mx + c
Coordinate grid with the line y = mx + c
  1. 1Translate the problem into precise notation, identify known quantities and constraints, and choose a representation that exposes the mathematical structure.
  2. 2Select the governing theorem, definition or model; state why its conditions hold before substituting or transforming any expression.
  3. 3Carry out the algebra, geometry or computation in justified stages, retaining exact forms and tracking domains, signs, dimensions and parameters.
  4. 4Verify the result through substitution, differentiation, an independent representation or a limiting case. The checked conclusion is (0, 0).

Answer: (0, 0)

Concept mastery

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Practice

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Common mistake. A common error in fundamental identities is to apply a familiar formula without confirming its hypotheses, domain or orientation. This can create an algebraically polished answer that is mathematically invalid. State the controlling condition first, preserve exact notation, and test the result against the original model rather than only the final line.

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