The Trigonographer’s Spotlight

• Angle Sum and Difference for Sine and Cosine
  \( \begin{align}
  \sin(\alpha\pm\beta) &= \sin\alpha \cos\beta \pm \cos\alpha \sin\beta \\
  \cos(\alpha\pm\beta) &= \cos\alpha \cos\beta \mp \sin\alpha \sin\beta
  \end{align}\)
• Combining Sine and Cosine
  \(
  \begin{align}
  p \sin\theta + q \cos\theta &= r \sin\left( \theta + \phi \right) \\
  p \cos\theta \,- q \sin\theta &= r \cos\left( \theta + \phi \right)
  \end{align}\)
• Exponential Forms of Hyperbolic Sine and Cosine
  \(
  \begin{align}
  2 \sinh u &= e^{u} -\,e^{-u} \\
  2 \cosh u &= e^{u} + e^{-u}
  \end{align}\)

• COEXIST
• Half-Angle Identities in a Triangle
• The Inscribed Angle Theorem without the Triangle Angle-Sum Theorem
• Pi by Ramanujan
• The Law of Cosines
• A Divided Angle Tangent Inequality
• Inradius and Circumradius of a Right Triangle
• Chicken-Taught Pi
• Euclid + Cupid
• Happy Hallowe’en
• Trigonography Challenge: Series for Cotangent and Cosecant
• Involute Zig-Zag: Power Series for Secant and Tangent
• Chaikovsky’s Involute Pinwheel: Power Series for Sine and Cosine
• Angles and Tangents in Arithmetic Progression
• Two Circles Puzzle
• Exponential Forms of Hyperbolic Sine and Cosine
• Special Angles are Golden, II
• Sum of Sines, Sum of Cosines, Sum of Angles
• (Nearly-)Pi from Square Roots
• Special Angles are Golden
• Compound Hypotenuse
• Trig Sums with Angles in Arithmetic Progression
• Three Squares Puzzle
• A Ratio of Trig Sums
• Cartographer’s Tangent Formulas
• An Arctangent Identity
• A Reciprocal Sum Identity
• Combining Sine and Cosine
• Angle Sum and Difference for Sine and Cosine
• Pythagoras in Disguise
• Half-Angle Differences for Triangles
• A Cotangent Identity for Triangles