\(\sin\left(\theta+\dfrac{\pi}{6}\right)\) \(=\dfrac{\sqrt{3}}{2}\sin\theta+\dfrac{1}{2}\cos\theta\)
\(\sin\left(\theta+\dfrac{\pi}{4}\right)\) \(=\dfrac{\sqrt{2}}{2}\sin\theta+\dfrac{\sqrt{2}}{2}\cos\theta\)
\(\sin\left(\theta+\dfrac{\pi}{3}\right)\) \(=\dfrac{1}{2}\sin\theta+\dfrac{\sqrt{3}}{2}\cos\theta\)

\(\sin\left(\theta-\dfrac{\pi}{6}\right)\) \(=\dfrac{\sqrt{3}}{2}\sin\theta-\dfrac{1}{2}\cos\theta\)
\(\sin\left(\theta-\dfrac{\pi}{4}\right)\) \(=\dfrac{\sqrt{2}}{2}\sin\theta-\dfrac{\sqrt{2}}{2}\cos\theta\)
\(\sin\left(\theta-\dfrac{\pi}{3}\right)\) \(=\dfrac{1}{2}\sin\theta-\dfrac{\sqrt{3}}{2}\cos\theta\)

\(\cos\left(\theta+\dfrac{\pi}{6}\right)\) \(=\dfrac{\sqrt{3}}{2}\cos\theta-\dfrac{1}{2}\sin\theta\)
\(\cos\left(\theta+\dfrac{\pi}{4}\right)\) \(=\dfrac{\sqrt{2}}{2}\cos\theta-\dfrac{\sqrt{2}}{2}\sin\theta\)
\(\cos\left(\theta+\dfrac{\pi}{3}\right)\) \(=\dfrac{1}{2}\cos\theta-\dfrac{\sqrt{3}}{2}\sin\theta\)

\(\cos\left(\theta-\dfrac{\pi}{6}\right)\) \(=\dfrac{\sqrt{3}}{2}\cos\theta+\dfrac{1}{2}\sin\theta\)
\(\cos\left(\theta-\dfrac{\pi}{4}\right)\) \(=\dfrac{\sqrt{2}}{2}\cos\theta+\dfrac{\sqrt{2}}{2}\sin\theta\)
\(\cos\left(\theta-\dfrac{\pi}{3}\right)\) \(=\dfrac{1}{2}\cos\theta+\dfrac{\sqrt{3}}{2}\sin\theta\)

\(\sin\left(45^\circ+30^\circ\right)\) \(=\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
\(\cos\left(45^\circ+30^\circ\right)\) \(=\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
\(\sin\left(60^\circ+45^\circ\right)\) \(=\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
\(\cos\left(60^\circ+45^\circ\right)\) \(=\dfrac{-\sqrt{6}+\sqrt{2}}{4}\)
\(\sin\left(60^\circ-45^\circ\right)\) \(=\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
\(\cos\left(60^\circ-45^\circ\right)\) \(=\dfrac{\sqrt{6}+\sqrt{2}}{4}\)

\(\tan\left(45^\circ+30^\circ\right)\) \(=2+\sqrt{3}\)
\(\tan\left(60^\circ+45^\circ\right)\) \(=-2-\sqrt{3}\)
\(\tan\left(60^\circ-45^\circ\right)\) \(=2-\sqrt{3}\)