Pertanyaan

8. Buktikan bahwa: c. sin 2 θ = 1 + tan 2 θ 2 tan θ

8. Buktikan bahwa:

c. $sin 2 θ = \frac{2 tan θ}{1 + tan ^{2} θ}$

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Klaim

I. Ridha

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

terbukti jika sin 2 θ = 1 + tan 2 θ 2 tan θ .

terbukti jika

sin 2 θ = \frac{2 tan θ}{1 + tan ^{2} θ}

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Pembahasan

Ingat kembali: rumussinus untuk sudut ganda: sin 2 α = 2 ⋅ sin α ⋅ cos α identitas trigonometri: tan 2 α + 1 = se c 2 α cos α = sec α 1 Oleh karena itu,dapat diperoleh: sin 2 θ = = = = = = = = = = 2 ⋅ sin θ ⋅ cos θ ( 2 ⋅ sin θ ⋅ cos θ ) ⋅ 1 ( 2 ⋅ sin θ ⋅ cos θ ) ⋅ ( c o s 2 x 1 ⋅ 1 c o s 2 x ) ( c o s 2 x 2 ⋅ s i n θ ⋅ c o s θ ) ⋅ cos 2 x ( c o s 2 x c o s θ 2 ⋅ s i n θ ⋅ c o s θ 1 ) ⋅ cos θ ⋅ cos θ ( c o s θ 2 ⋅ s i n θ ⋅ 1 ) ⋅ sec x 1 ⋅ sec x 1 ( 2 ⋅ c o s θ s i n θ ) ⋅ se c 2 x 1 ( 2 ⋅ tan θ ) ⋅ 1 + t a n 2 θ 1 1 + t a n 2 θ 2 ⋅ t a n θ ⋅ 1 1 + t a n 2 θ 2 t a n θ Dengan demikian, terbukti jika sin 2 θ = 1 + tan 2 θ 2 tan θ .

Ingat kembali:

rumus sinus untuk sudut ganda: $sin 2 α = 2 \cdot sin α \cdot cos α$
identitas trigonometri: $tan^{2} α + 1 = se c^{2} α$
$cos α = \frac{1}{sec α}$

Oleh karena itu, dapat diperoleh:

$sin 2 θ = = = = = = = = = = 2 \cdot sin θ \cdot cos θ (2 \cdot sin θ \cdot cos θ) \cdot 1 (2 \cdot sin θ \cdot cos θ) \cdot (\frac{1}{c o s ^{2} x} \cdot \frac{c o s ^{2} x}{1}) (\frac{2 \cdot s i n θ \cdot c o s θ}{c o s ^{2} x}) \cdot cos^{2} x (\frac{2 \cdot s i n θ \cdot c o s θ ^{1}}{c o s ^{2} x ^{c o s θ}}) \cdot cos θ \cdot cos θ (\frac{2 \cdot s i n θ \cdot 1}{c o s θ}) \cdot \frac{1}{sec x} \cdot \frac{1}{sec x} (2 \cdot \frac{s i n θ}{c o s θ}) \cdot \frac{1}{se c ^{2} x} (2 \cdot tan θ) \cdot \frac{1}{1 + t a n ^{2} θ} \frac{2 \cdot t a n θ \cdot 1}{1 + t a n ^{2} θ} \frac{2 t a n θ}{1 + t a n ^{2} θ}$