Tentukan nilai-nilai x yang memenuhi persamaan berikut untuk 0° ≤ x ≤ 360°. 3 sin x + 1 = 2 csc x

<p>Jawab: <strong><u>x = 135,5°, x=44,5°, dan x=x = 270°.</u></strong></p><p>Ingat!</p><ul><li>cosec 𝞪 = 1/sin𝞪.</li><li>sin x = sin p, maka x = p+k.360° atau x = (180°-p)+k.360°</li></ul><p>Dengan menggunakan konsep di atas, diperoleh</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;3 sin x + 1 = 2 csc x</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3 sinx + 1 = 2(1/sinx)</p><p>&nbsp; &nbsp; &nbsp; 3sin²x+sin x = 2</p><p>3sin²x+sin x - 2 = 0</p><p>Misal p = sinx, maka</p><p>3sin²x+sin x - 2 = 0</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3p²+p-2 = 0</p><p>&nbsp; &nbsp; &nbsp;3p²+3p-2p-2 = 0</p><p>3p(p+1)-2(p+1) = 0</p><p>&nbsp; &nbsp; &nbsp; &nbsp;(3p-2)(p+1) = 0</p><p>3p-2 = 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; atau &nbsp; &nbsp; &nbsp; p+1 = 0</p><p>&nbsp; &nbsp; &nbsp; &nbsp;p = 2/3 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; p = -1</p><p>Untuk p=2/3.</p><p>sin x = 2/3&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; x ≈ &nbsp;44,5° +k.360°(menggunakan kalkulator)</p><p>k = 0 maka x = 44,5°</p><p>k= 1 maka x = 404,5° (tidak memenuhi).</p><p>sin x = 2/3&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; x ≈ &nbsp;(180°-44,4°) +k.360°</p><p>&nbsp; &nbsp; &nbsp; &nbsp; x ≈ &nbsp;135,5° +k.360°</p><p>k = 0 maka x = 135,5°</p><p>k= 1 maka x = 495,5° (tidak memenuhi).</p><p>Untuk p = -1.</p><p>sin x = -1</p><p>&nbsp; &nbsp; &nbsp; &nbsp;x = &nbsp;270°+k.360°</p><p>k=0 maka &nbsp; x = 270°</p><p>k= 1 maka x = 630° (tidak memenuhi).</p><p>sin x = -1</p><p>&nbsp; &nbsp; &nbsp; &nbsp;x = &nbsp;(180°-270°)+k.360°</p><p>&nbsp; &nbsp; &nbsp; &nbsp;x = -90°+k.360°&nbsp;</p><p>k=0 maka &nbsp; x = -90°(tidak memenuhi)</p><p>k= 1 maka x = 270°.</p><p>&nbsp;</p><p>Jadi, jawaban yang benar adalah <strong><u>x = 135,5°, x=44,5°, dan x=x = 270°.</u></strong></p>