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• An identity is a mathematical statement equating one quantity to another; trigonometric identities simplify expressions using sine and cosine ratios, facilitating equation-solving and identity-proving.

Quotient Identity

• The quotient identity is expressed as $ \tan \theta = \frac{\sin \theta}{\cos \theta} $.
• This identity is derived from $ \tan \theta = \frac{y}{x} $, where $y$ is the opposite side and $x$ is the adjacent side in a right triangle, by multiplying by $ \frac{r}{r} $ and substituting with $ \sin \theta = \frac{y}{r} $ and $ \cos \theta = \frac{x}{r} $.
• $ \tan \theta $ is undefined when $ \cos \theta = 0 $, which occurs when $ \theta = k \times 90^\circ $ where $k$ is an odd integer.

Square Identity (Pythagorean Identity)

• The square identity is $ \sin^2 \theta + \cos^2 \theta = 1 $.
• The derivation uses the Pythagorean theorem, in a right triangle, $ \sin^2 \theta + \cos^2 \theta = (\frac{y}{r})^2 + (\frac{x}{r})^2 = \frac{y^2 + x^2}{r^2} = \frac{r^2}{r^2} = 1 $.
• Alternate forms of this identity include $ \sin^2 \theta = 1 - \cos^2 \theta $ and $ \cos^2 \theta = 1 - \sin^2 \theta $.

Summary of Fundamental Identities

• $ \tan \theta = \frac{\sin \theta}{\cos \theta} $ where undefined when $ \cos \theta = 0 $.
• $ \sin^2 \theta + \cos^2 \theta = 1 $

Reduction Formulas

• Trigonometric functions with arguments $90^\circ \pm \theta$, $180^\circ \pm \theta$, $360^\circ \pm \theta$, or $ -\theta $ can be expressed in terms of $ \theta $.

Reduction Formulas for $90^\circ \pm \theta$

• $ \sin(90^\circ - \theta) = \cos \theta $ and $ \cos(90^\circ - \theta) = \sin \theta $.
• $ \sin(90^\circ + \theta) = \cos \theta $ and $ \cos(90^\circ + \theta) = -\sin \theta $.
• Derivation example: Reflecting point $ P(\sqrt{3}, 1) $ over $ y = x $ gives $ P'(1, \sqrt{3}) $, demonstrating $ \sin(90^\circ - \theta) = \frac{\sqrt{3}}{2} = \cos 30^\circ = \cos \theta $.

Reduction Formulas for $180^\circ \pm \theta$

• $ \sin(180^\circ - \theta) = \sin \theta $, $ \cos(180^\circ - \theta) = -\cos \theta $, and $ \tan(180^\circ - \theta) = -\tan \theta $.
• $ \sin(180^\circ + \theta) = -\sin \theta $, $ \cos(180^\circ + \theta) = -\cos \theta $, and $ \tan(180^\circ + \theta) = \tan \theta $.
• Derivation example: Reflecting point $ P(\sqrt{3}, 1) $ over the y-axis gives $ P'(-\sqrt{3}, 1) $, demonstrating $ \cos(180^\circ - \theta) = \frac{-\sqrt{3}}{2} = -\cos 30^\circ = -\cos \theta $.

Reduction Formulas for $360^\circ \pm \theta$ and $ -\theta $

• $ \sin(360^\circ - \theta) = -\sin \theta $, $ \cos(360^\circ - \theta) = \cos \theta $, and $ \tan(360^\circ - \theta) = -\tan \theta $.
• $ \sin(-\theta) = -\sin \theta $, $ \cos(-\theta) = \cos \theta $, and $ \tan(-\theta) = -\tan \theta $.
• $ \sin(360^\circ + \theta) = \sin \theta $, $ \cos(360^\circ + \theta) = \cos \theta $, and $ \tan(360^\circ + \theta) = \tan \theta $.

Summary of Reduction Formulas

• $ \sin(90^\circ \pm \theta) = \pm \cos \theta $
• $ \cos(90^\circ \pm \theta) = \mp \sin \theta $
• $ \sin(180^\circ \pm \theta) = \pm \sin \theta $
• $ \cos(180^\circ \pm \theta) = -\cos \theta $
• $ \tan(180^\circ \pm \theta) = \pm \tan \theta $
• $ \sin(360^\circ \pm \theta) = \mp \sin \theta $
• $ \cos(360^\circ \pm \theta) = \pm \cos \theta $
• $ \tan(360^\circ \pm \theta) = \mp \tan \theta $
• $ \sin(-\theta) = -\sin \theta $
• $ \cos(-\theta) = \cos \theta $
• $ \tan(-\theta) = -\tan \theta $

Trigonometric Equations

• General solutions account for periodicity in trigonometric functions.
• If $ \sin \theta = x $, then $ \theta = \sin^{-1} x + k \cdot 360^\circ $ or $ \theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ $ where $ k \in \mathbb{Z} $.
• If $ \cos \theta = x $, then $ \theta = \cos^{-1} x + k \cdot 360^\circ $ or $ \theta = 360^\circ - \cos^{-1} x + k \cdot 360^\circ $ where $ k \in \mathbb{Z} $.
• If $ \tan \theta = x $, then $ \theta = \tan^{-1} x + k \cdot 180^\circ $ where $ k \in \mathbb{Z} $.

Solving Method

• Simplify the equation using identities.
• Find reference angle using the inverse trigonometric functions.
• Use the CAST diagram to determine quadrants.
  • Quadrant I ($0^\circ - 90^\circ $): All positive
  • Quadrant II ($90^\circ - 180^\circ $): Sine positive
  • Quadrant III ($180^\circ - 270^\circ $): Tangent positive
  • Quadrant IV ($270^\circ - 360^\circ $): Cosine positive
• Apply reduction formulas and add multiples of the period.
• Verify with a calculator.

Quadratic Trigonometric Equations

• Simplify the equation and find the reference angle.
• Use the CAST diagram to identify quadrants.
• Substitute values to find solutions.
• Verify solutions.

Area Rule

• The area of a triangle $ \triangle ABC $ can be found using the formulas: $ \text{Area} = \frac{1}{2} bc \sin A = \frac{1}{2} ab \sin C = \frac{1}{2} ac \sin B $.

Sine Rule

• For any triangle $ \triangle ABC $, the sine rule states: $ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $.

Cosine Rule

• For any triangle $ \triangle ABC $, the cosine rules are:
  • $ a^2 = b^2 + c^2 - 2bc \cos A $
  • $ b^2 = a^2 + c^2 - 2ac \cos B $
  • $ c^2 = a^2 + b^2 - 2ab \cos C $

Summary of Usage

• Area Rule: Use when no height is given.
• Sine Rule: Use when given two angles and one side, or two sides and a non-included angle.
• Cosine Rule: Use when given two sides and the included angle, or three sides.