Quick Recap — Values, Equations & Extremes

  • sin⁡75∘=3+122\sin75^\circ=\tfrac{\sqrt3+1}{2\sqrt2}, cos⁡75∘=3−122\cos75^\circ=\tfrac{\sqrt3-1}{2\sqrt2}, tan⁡75∘=2+3\tan75^\circ=2+\sqrt3, tan⁡15∘=2−3\tan15^\circ=2-\sqrt3.
  • General solutions: sin⁡θ=0⇒θ=nπ\sin\theta=0\Rightarrow\theta=n\pi; cos⁡θ=0⇒θ=(2n+1)π2\cos\theta=0\Rightarrow\theta=(2n+1)\tfrac{\pi}{2}; tan⁡θ=tan⁡α⇒θ=nπ+α\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alpha.
  • Range of asin⁡θ+bcos⁡θa\sin\theta+b\cos\theta is [−a2+b2, a2+b2]\big[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\big].
  • sin⁡18∘=5−14\sin18^\circ=\tfrac{\sqrt5-1}{4}, cos⁡36∘=5+14\cos36^\circ=\tfrac{\sqrt5+1}{4}.