how to find the trigonometric ratios of 67.5° (SINE, COSINE, TAN) without calculator?

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03-11-2017
Mathematics

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how to find the trigonometric ratios of 67.5° (SINE, COSINE, TAN) without calculator?

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Rod44 Rod44 · Answer 1 · 2017-11-03T05:04:57+01:00

Rod44 Rod44

03-11-2017

90-67.5=22.5=45/2.
TAN45=1 and TAN45=2TAN22.5/(1-TAN²22.5)=1
Let t=TAN22.5: 2t=1-t², t²+2t=1, t²+2t+1=2, (t+1)²=2, t=-1±√2. TAN22.5=√2-1.
TAN67.5=COT22.5=1/(√2-1)=√2+1.
Consider a right triangle with sides √2+1, 1 and √((√2+1)²+1)=√(4+2√2).
SIN67.5=(√2+1)/√(4+2√2); COS67.5=1/√(4+2√2).

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