Trigonometry Basics

Circle and Arc Length

• Radius (r): Distance from the center to any point on the circle.
• Arc Length (s): Distance along the circle's circumference.
  • Formula: ( s = \theta \times r )
  • ( \theta ) should be in radians.
  • Units for arc length match units for the radius.
• Conversions:
  • ( 360^{\circ} = 2\pi ) radians
  • Conversion videos available online.

Area of a Sector

• Formula 1: ( \text{Area} = \frac{1}{2} \times \theta \times r^2 )
  • ( \theta ) in radians.
• Formula 2: ( \text{Area} = \frac{\theta}{360^{\circ}} \times \pi \times r^2 )
  • ( \theta ) in degrees.

Right Triangle Trigonometry

• Components:
  • Opposite Side: Opposite the angle ( \theta ).
  • Adjacent Side: Next to the angle ( \theta ).
  • Hypotenuse: Longest side.
• SOHCAHTOA Mnemonic:
  • Sine (SOH): ( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} )
  • Cosine (CAH): ( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
  • Tangent (TOA): ( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} )

Reciprocal Identities

• Secant (sec): ( \sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} )
• Cosecant (csc): ( \csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}} )
• Cotangent (cot): ( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}} )

Quotient Identities

• ( \tan \theta = \frac{\sin \theta}{\cos \theta} )
• ( \cot \theta = \frac{\cos \theta}{\sin \theta} )

Pythagorean Identities

• ( \sin^2 \theta + \cos^2 \theta = 1 )
• ( 1 + \tan^2 \theta = \sec^2 \theta )
• ( 1 + \cot^2 \theta = \csc^2 \theta )

Even-Odd Identities

• Odd Functions:
  • ( \sin(-\theta) = -\sin \theta )
  • ( \tan(-\theta) = -\tan \theta )
  • ( \csc(-\theta) = -\csc \theta )
  • ( \cot(-\theta) = -\cot \theta )
• Even Functions:
  • ( \cos(-\theta) = \cos \theta )
  • ( \sec(-\theta) = \sec \theta )

Co-Function Identities

• Complementary Angles:
  • ( \sin(90^{\circ} - \theta) = \cos \theta )
  • ( \tan(90^{\circ} - \theta) = \cot \theta )
  • ( \sec(90^{\circ} - \theta) = \csc \theta )
  • ( \csc(90^{\circ} - \theta) = \sec \theta )

Double Angle Formulas

• Sine:
  • ( \sin 2\theta = 2 \sin \theta \cos \theta )
  • Alternative: ( \sin 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta} )
• Cosine:
  • ( \cos 2\theta = \cos^2 \theta - \sin^2 \theta )
  • Alternative forms: ( \cos 2\theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta )
• Tangent:
  • ( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} )

Additional Resources

• Formula Sheet: Available for more advanced formulas (half-angle, triple-angle, power reducing, etc.).
• Further Study: Check online resources or the provided formula sheet link for additional identities and formulas.