F. 3 數學問題 (Trigonmetric Relations) 急!!!

2007-04-09 3:15 am
Prove the following trigonometric identities.

1. 1 + tan^2 Θ ≡ 1 / cos^2 Θ

2. sin Θ + cosΘ / tanΘ ≡ 1 / sinΘ


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回答 (3)

2007-04-09 3:22 am
✔ 最佳答案
1. 1 + tan^2 Θ = 1 / cos^2 Θ

LHS

=1 + tan^2 Θ

=sec^2 Θ

=1/cos^2 Θ

=RHS


or


LHS

=1+ tan^2 Θ

=1+(sin^2 Θ/cos^2 Θ)

=(cos^2 Θ+sin^2 Θ)/cos^2 Θ

=1/cos^2 Θ

=RHS

-----------------------------------------------------
2. sin Θ + cosΘ/ tanΘ = 1 / sinΘ

LHS

=sin Θ + cosΘ / tanΘ

=sin Θ+cosΘ/(sin Θ/cos Θ)

=sin Θ+cos^2Θ/sin Θ

=(sin^2 Θ+cos^2Θ)/sin Θ

=1/sin Θ

=RHS
2007-04-09 3:25 am
1.
1 + tan^2 Θ
= 1 + sin^2Θ / cos^2Θ
= ( cos^2Θ + sin^2Θ ) / cos^2Θ
= 1 / cos^2 Θ ****cos^2Θ + sin^2Θ=1******

2.
sin Θ + cosΘ / tanΘ
= sin Θ + cos^2Θ / sin Θ
= ( sin^2Θ + cos^2Θ ) / sin Θ
= 1 / sinΘ ****cos^2Θ + sin^2Θ=1******
參考: 幫到你嗎?
2007-04-09 3:22 am
1)1 + tan^2 Θ = 1 + sin^2 Θ / cos^2 Θ
      = (cos^2 Θ + sin^2 Θ) / cos^2 Θ <=[通分母]
      = 1 / cos^2 Θ // <=[sin^2 Θ + cos^2 Θ) =1]

2)sin Θ + cos Θ / tan Θ = sin Θ + cos Θ / (sin Θ / cos Θ)
          = sin Θ + cos^2 Θ / sin Θ <=[除分母=乘分母倒數]
          = (sin^2 Θ + cos^2 Θ) / sin Θ <=[通分母]
          = 1 / sin Θ // <=[sin^2 Θ + cos^2 Θ) =1]
參考: by My Maths


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