nilai sin 18° dan cos 18°

<p>Jawaban: sin 18° = [-1 + √5]/4 dan cos 18° = [√(10+2√5)]/4.</p><p>&nbsp;</p><p>Ingat.</p><p>Pada segitiga siku-siku berlaku:</p><p>sin x = depan/miring</p><p>cos x = samping/miring</p><p>&nbsp;</p><p>Ingat juga</p><p>sin 2x= 2 sin x cos x</p><p>cos 2x=1 - 2 sin² x</p><p>sin (90°- x)=cos x</p><p>cos(x+y) = cos x cos y - sin x sin y</p><p>nilai sin dan cos di kudaran I bernilai positif.</p><p>&nbsp;</p><p>Ditanya: nilai sin 18° dan cos 18°?</p><p>&nbsp;</p><p>Misalkan x=18°</p><p>5x=90°</p><p>2x+3x=90°</p><p>2x=90°-3x</p><p>sin 2x=sin (90°-3x)</p><p>2 sin x cos x = cos 3x</p><p>2 sin x cos x = cos (2x+x)</p><p>2 sin x cos x = cos 2x cos x - sin 2x sin x</p><p>2 sin x cos x = (1 - 2 sin² x)cos x - (2 sin x cos x) sin x</p><p>2 sin x cos x = (1 - 2 sin² x) cos x - 2 sin² x cos x</p><p>2 sin x <strong>cos x</strong> = <strong>cos x</strong> (1 - 2 sin² x - 2 sin² x)&nbsp;</p><p>2 sin x = 1 - 4 sin² x</p><p>4 sin² x + 2 sin x - 1 = 0</p><p>misal sin x = p</p><p>4p² + 2p - 1 = 0</p><p>&nbsp;</p><p>Akan dicari nilai p dengan rumus kuadratik</p><p>p = [-b ±√(b²-4ac)]/2a</p><p>p = [-2 ±√(2²-4.4.(-1))]/2.4</p><p>p = [-2 ±√(4+16)]/8</p><p>p = [-2 ±√20]/8</p><p>p = [-2 ±2√5]/8</p><p>p1= [-1 + √5]/4</p><p>p2= [-1 - √5]/4</p><p>&nbsp;</p><p>sin x = [-1 + √5]/4 atau sin x = [-1 - √5]/4</p><p>x= 18° maka x berada di kuadran I, sehingga sin 18°= [-1 + √5]/4.</p><p>&nbsp;</p><p>Nilai cos 18° akan dicari menggunakan pythagoras.</p><p>sin 18°= [-1 + √5]/4</p><p>depan = -1 + √5</p><p>miring = 4</p><p>samping = √[4²-(-1 + √5)²]</p><p>= √[16-(6-2√5)]</p><p>= √(10+2√5)</p><p>&nbsp;</p><p>cos 18° = samping/miring</p><p>cos 18° = [√(10+2√5)]/4</p><p>&nbsp;</p><p>Jadi, nilai sin 18° = [-1 + √5]/4 dan cos 18° = [√(10+2√5)]/4.</p>