Pertanyaan

Hitunglah b. cos 2 1 5 ∘ − cos 2 7 5 ∘

Hitunglah

b. $cos^{2} 1 5^{\circ} - cos^{2} 7 5^{\circ}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

diperoleh cos 2 ( 1 5 ∘ ) − cos 2 ( 7 5 ∘ ) = 2 1 3

diperoleh

cos^{2} (1 5^{\circ}) - cos^{2} (7 5^{\circ}) = \frac{1}{2} 3

Pembahasan

Akan dicari cos 2 ( 1 5 ∘ ) − cos 2 ( 7 5 ∘ ) Ingat bahwa a 2 − b 2 = ( a + b ) ( a − b ) cos A − cos B = − 2 sin 2 1 ( A + B ) sin 2 1 ( A − B ) cos A + cos B = 2 cos 2 1 ( A + B ) cos 2 1 ( A − B ) Diperoleh perhitungan = = = = = = = = = cos 2 ( 1 5 ∘ ) − cos 2 ( 7 5 ∘ ) ( cos 1 5 ∘ + cos 7 5 ∘ ) ( cos 1 5 ∘ − cos 7 5 ∘ ) ( 2 cos 2 1 ( 1 5 ∘ + 7 5 ∘ ) cos 2 1 ( 1 5 ∘ − 7 5 ∘ ) ) ( − 2 sin 2 1 ( 1 5 ∘ + 7 5 ∘ ) sin 2 1 ( 1 5 ∘ − 7 5 ∘ ) ) ( 2 cos 2 1 ( 9 0 ∘ ) cos 2 1 ( − 6 0 ∘ ) ) ( − 2 sin 2 1 ( 9 0 ∘ ) sin 2 1 ( − 6 0 ∘ ) ) ( 2 cos ( 4 5 ∘ ) cos ( − 3 0 ∘ ) ) ( − 2 sin ( 4 5 ∘ ) sin ( − 3 0 ∘ ) ) ( 2 ⋅ 2 1 2 ⋅ 2 1 3 ) ( − 2 ⋅ 2 1 2 ⋅ − 2 1 ) 2 6 ⋅ 2 1 2 4 1 12 4 1 ⋅ 2 3 2 1 3 Dengan demikian, diperoleh cos 2 ( 1 5 ∘ ) − cos 2 ( 7 5 ∘ ) = 2 1 3

Akan dicari $cos^{2} (1 5^{\circ}) - cos^{2} (7 5^{\circ})$

Ingat bahwa

$a^{2} - b^{2} = (a + b) (a - b) cos A - cos B = - 2 sin \frac{1}{2} (A + B) sin \frac{1}{2} (A - B) cos A + cos B = 2 cos \frac{1}{2} (A + B) cos \frac{1}{2} (A - B)$

Diperoleh perhitungan

$= = = = = = = = = cos^{2} (1 5^{\circ}) - cos^{2} (7 5^{\circ}) (cos 1 5^{\circ} + cos 7 5^{\circ}) (cos 1 5^{\circ} - cos 7 5^{\circ}) (2 cos \frac{1}{2} (1 5^{\circ} + 7 5^{\circ}) cos \frac{1}{2} (1 5^{\circ} - 7 5^{\circ})) (- 2 sin \frac{1}{2} (1 5^{\circ} + 7 5^{\circ}) sin \frac{1}{2} (1 5^{\circ} - 7 5^{\circ})) (2 cos \frac{1}{2} (9 0^{\circ}) cos \frac{1}{2} (- 6 0^{\circ})) (- 2 sin \frac{1}{2} (9 0^{\circ}) sin \frac{1}{2} (- 6 0^{\circ})) (2 cos (4 5^{\circ}) cos (- 3 0^{\circ})) (- 2 sin (4 5^{\circ}) sin (- 3 0^{\circ})) (2 \cdot \frac{1}{2} 2 \cdot \frac{1}{2} 3) (- 2 \cdot \frac{1}{2} 2 \cdot - \frac{1}{2}) \frac{6}{2} \cdot \frac{1}{2} 2 \frac{1}{4} 12 \frac{1}{4} \cdot 23 \frac{1}{2} 3$