IB Mathematics AA Relationships in trigonometric functions Study Notes

IB Mathematics AA Relationships in trigonometric functions Study Notes

IB Mathematics AA Relationships in trigonometric functions Study Notes

IB Mathematics AA Relationships in trigonometric functions Study Notes Offer a clear explanation of Relationships in trigonometric functions, including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Relationships in trigonometric functions.

Relationships Between Trigonometric Functions and Their Symmetry

Relationships Between Trigonometric Functions and Their Symmetry

Trigonometric functions have specific symmetry and quadrant properties reflected in their graphs:

Sine function, \( \sin x \):

  • Odd function: \( \sin(-x) = -\sin x \)
  • Symmetry: Graph is symmetric about the origin.
  • Identity: \( \sin(\pi – x) = \sin x \)

Cosine function, \( \cos x \):

  • Even function: \( \cos(-x) = \cos x \)
  • Symmetry: Graph is symmetric about the y-axis.
  • Identity: \( \cos(\pi – x) = -\cos x \)

Tangent function, \( \tan x \):

  • Odd function: \( \tan(-x) = -\tan x \)
  • Symmetry: Graph is symmetric about the origin.
  • Identity: \( \tan(\pi – x) = -\tan x \)

Cosecant function, \( \csc x \):

  • Odd function: \( \csc(-x) = -\csc x \)
  • Symmetry: Graph is symmetric about the origin.
  • Identity: \( \csc(\pi – x) = \csc x \)

Secant function, \( \sec x \):

  • Even function: \( \sec(-x) = \sec x \)
  • Symmetry: Graph is symmetric about the y-axis.
  • Identity: \( \sec(\pi – x) = -\sec x \)

Cotangent function, \( \cot x \):

  • Odd function: \( \cot(-x) = -\cot x \)
  • Symmetry: Graph is symmetric about the origin.
  • Identity: \( \cot(\pi – x) = -\cot x \)

Example

Given \( \theta = 30^\circ \) or \( \theta = \frac{\pi}{6} \), use symmetry and π − x identities to find:

  • \( \sin(-\theta) \)
  • \( \cos(-\theta) \)
  • \( \tan(-\theta) \)
  • \( \sin(\pi – \theta) \)
  • \( \cos(\pi – \theta) \)
  • \( \tan(\pi – \theta) \)
▶️Answer/Explanation

\( \sin(-\theta) \):
Since sin is odd: \( \sin(-\theta) = -\sin \theta = -\frac{1}{2} \)

\( \cos(-\theta) \):
Since cos is even: \( \cos(-\theta) = \cos \theta = \frac{\sqrt{3}}{2} \)

\( \tan(-\theta) \):
Since tan is odd: \( \tan(-\theta) = -\tan \theta = -\frac{1}{\sqrt{3}} \)

\( \sin(\pi – \theta) \):
\( \sin(\pi – \theta) = \sin \theta = \frac{1}{2} \)

\( \cos(\pi – \theta) \):
\( \cos(\pi – \theta) = -\cos \theta = -\frac{\sqrt{3}}{2} \)

\( \tan(\pi – \theta) \):
\( \tan(\pi – \theta) = -\tan \theta = -\frac{1}{\sqrt{3}} \)

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