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
- 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
\(c^2 \left( \cot A + \cot C \right) = b^2 \left( \cot A + \cot B \right) \) - A Ratio of Trig Sums
Odom’s construction of the golden ratio in an equilateral triangle has counterparts in a square and regular hexagon.