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
  \(
  \begin{align}
  2 \sinh u &= e^{u} -\,e^{-u} \\
  2 \cosh u &= e^{u} + e^{-u}
  \end{align}\)

• Special Angles are Golden, II
  Đào’s constructions of the golden ratio in an equilateral triangle and a square have a counterpart in the regular hexagon.
  
  
• Sum of Sines, Sum of Cosines, Sum of Angles
  \(\left.\begin{align}
  \sin\theta + \sin\phi &= a \\[4pt]
  \cos\theta + \cos\phi &= b
  \end{align}\quad\right\} \quad\Longrightarrow\quad\sin\left(\theta+\phi\right) = \dfrac{2ab}{a^2+b^2}\)
• (Nearly-)Pi from Square Roots
  \( \sqrt{2} + \sqrt{3} \approx \pi \)
• Special Angles are Golden
  Odom’s construction of the golden ratio in an equilateral triangle has counterparts in a square and regular hexagon.

• Compound Hypotenuse
  \(\begin{align}
  &\left(\;\sin 2\alpha + \sin 2\beta + 2 \sin(\alpha+\beta) \;\right)^2 \\
  + \;\;&\left(\;\cos 2\alpha + \cos 2\beta + 2 \cos(\alpha+\beta) \;\right)^2 \\
  = \;\;&\left(\;2 \left(\; 1 + \cos(\alpha-\beta) \;\right) \;\right)^2\end{align}\)
• (Nearly-)Pi from Square Roots
  \( \sqrt{2} + \sqrt{3} \approx \pi \)
• Trig Sums with Angles in Arithmetic Progression
  \(
  \begin{align}
  \sum_{k=0}^{n-1}\;\sin 2k\theta &=\frac{\sin n\theta}{\sin\theta}\;\sin(n-1)\theta \\
  \sum_{k=0}^{n-1}\;\cos 2k\theta &=\frac{\sin n\theta}{\sin\theta}\;\cos(n-1)\theta
  \end{align} \)
• Three Squares Puzzle
  \( \alpha + \beta + \gamma = \text{???} \)
• A Ratio of Trig Sums
  \( \dfrac{1+\cos\theta+\sin\theta}{1+\sin\theta-\cos\theta} \;=\; \dfrac{1+\cos\theta}{\sin\theta} \)
• Cartographer’s Tangent Formulas
  \( \sec\theta \pm \tan \theta = \tan\left( 45^\circ \pm \dfrac{\theta}{2} \right) \)
• An Arctangent Identity
  \( \operatorname{atan}\dfrac{x}{1} + \operatorname{atan}\dfrac{1-x}{1+x} \;=\; 45^\circ \)
• A Reciprocal Sum Identity
  \( x + \dfrac{1}{x} \;\geq\; 2 \)
• Combining Sine and Cosine
  \(
  \begin{align}
  p \sin\theta + q \cos\theta &= r \sin\left( \theta + \phi \right) \\
  p \cos\theta \,- q \sin\theta &= r \cos\left( \theta + \phi \right)
  \end{align}\)
• Angle Sum and Difference for Sine and Cosine
  \( \begin{align}
  \sin(\alpha\pm\beta) &= \sin\alpha \cos\beta \pm \cos\alpha \sin\beta \\
  \cos(\alpha\pm\beta) &= \cos\alpha \cos\beta \mp \sin\alpha \sin\beta
  \end{align}\)
• Pythagoras in Disguise
  \( \dfrac{1-\sin\theta}{\cos\theta} = \dfrac{1}{\sec\theta + \tan\theta} \)
• Half-Angle Differences for Triangles
  \(\left(a-b\right) \cos\frac{C}{2} = c \sin\frac{A-B}{2} \)
• A Cotangent Identity for Triangles
  \(c^2 \left( \cot A + \cot C \right) = b^2 \left( \cot A + \cot B \right) \)