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E6.3 Exact trigonometric values

Syllabus
0580–2028–2029
Topic
E6.3
Level
Extended

Know the exact trigonometric values

Exact trigonometric values use fractions and square roots instead of rounded calculator decimals. They let an expression be simplified without losing accuracy.

xx 0∘0^\circ 30∘30^\circ 45∘45^\circ 60∘60^\circ 90∘90^\circ
sin⁡x\sin x 00 12\frac12 22\frac{\sqrt2}{2} 32\frac{\sqrt3}{2} 11
cos⁡x\cos x 11 32\frac{\sqrt3}{2} 22\frac{\sqrt2}{2} 12\frac12 00
tan⁡x\tan x 00 33\frac{\sqrt3}{3} 11 3\sqrt3 not required

For 0∘,30∘,45∘,60∘,90∘0^\circ,30^\circ,45^\circ,60^\circ,90^\circ, the sine numerators follow 0,1,2,3,4\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4, all over 22. Cosine uses the same list in reverse. For the required tangent angles, use tan⁡x=sin⁡x/cos⁡x\tan x=\sin x/\cos x and simplify exactly.

Simplify 4.5cos⁡30∘3−2\dfrac{4.5\cos30^\circ}{\sqrt3}-2. Substitute cos⁡30∘=32\cos30^\circ=\dfrac{\sqrt3}{2}: 4.5(3/2)3−2=2.25−2=0.25=14\dfrac{4.5(\sqrt3/2)}{\sqrt3}-2=2.25-2=0.25=\dfrac14. The 3\sqrt3 factors cancel before any decimal approximation is needed.

Do not replace exact fractions or surds with rounded decimals unless the question asks for an approximation. Since cos⁡90∘=0\cos90^\circ=0, tan⁡90∘=sin⁡90∘/cos⁡90∘\tan90^\circ=\sin90^\circ/\cos90^\circ would divide by zero; this is why 90∘90^\circ is not in the required tangent list.