1. Cho α là góc mà tanα = 2. Tính:
P= sinα/ sin3α + 3cos3α
<=>sinα/cos3α / sin3α/cos3α
+ 3cos3α/cos3α
<=> sinα/cosα.
1/cos2α / tan3α + 3
<=> tanα.
(1 + tan2α) / tan3α + 3
<=> 2. [1 + (2)2] / 22 + 3
=> 10/11
2.
Giải
phương trình:
cos3x –
sin(2x – π/4) = 0
<=> –
sin(2x – π/4) = – cos3x
<=> sin(2x – π/4) = cos3x
<=> sin(2x – π/4) = sin(π/2 – 3x)
<=> 2x – π/4 = π/2 –
3x + k2π
2x – π/4 = π – π/2 – 3x + k2π
<=> 2x + 3x = π/2 + π/4 + k2π
2x + 3x = π – π/2 + π/4 + k2π
<=> 5x = 3π/4 + k2π
5x = 3π/4 + k2π
<=> x = 3π/20 + k2π/5
x = 3π/20 + k2π/5 (k ∈ Z)
3.
Cho
góc α ∈ (π/2 ; π) mà sin α/2 – cos α/2 = ½. Tính
sin2α
<=> (sin α/2 – cos α/2) = (1/2)2
<=> sin α/22
– 2sin α/2.cos α/2 + cosα/22 = ¼
<=> sin α/22
+ cos α/22 – 2sin α/2.cos α/2 = ¼
<=> (sin
α/2 + cos α/2)2 – 2sin α/2.cos α/2 = ¼
<=> 1 –
2sin α/2.cos α/2 = ¼ (α = α/2. α/2)
<=> 1 – sin
α = ¼
<=> – sin α
= ¼ - 1
<=> – sin α
= -3/4
=> sin α = ¾
sin2 α + cos2 α = 1
<=> (3/4)2
+ cos2 α = 1
<=> 9/16
+ cos2 α = 1
<=> cos2 α = 1 – 9/16
<=> cos2 α =
7/16
<=> cos α = +-√7/16
=> cos α =
-√7/16 (Vì điều kiện lấy từ π/2 ; π)
4.
Giải phương
trình √3sin2x + cos2x = sinx + √3cosx
Ta có √(√3)2 + 12 = 2
<=> √3/2.sin2x + ½.cos2x = 1/2.sinx + √3/2.cosx
<=> √3/2.sin2x + ½.cos2x = 1/2.sinx + √3/2.cosx
<=> cos√3/2.sin2x
+ sin½.cos2x = cos1/2.sinx + sin√3/2.cosx
<=> cos.π/6.sin2x + sin.π/6.cos2x = cos.π/3.sinx
+ sin.π/3cosx
<=> sin2x.cos.π/6 +.cos2x. sin.π/6 = sinx.cos.π/3 + cosx.
sin.π/3
<=> sin(π/6
+ 2x) = sin(π/3 + x)
<=> π/6 + 2x
= π/3 + x + k2π
π/6 + 2x
= π - π/3 + x + k2π
<=> 2x – x = π/3 – π/6 + k2π
2x –
x = π – π/3 – π/6 + k2π
<=> x = π/6 + k2π
x = π/2 + k2π (k ∈ Z)
5. Cho
góc α ∈ (π ; 3 π /2) mà cos α = -9/41. Tính tan(α + π/4)
<=> 1/cos2α = 1 +
tan2α
<=> 1/
(-9/41)2 = 1 + tan2α
<=> 1681/81 = 1 + tan2α
<=> 1681/81 – 1 = 1 + tan2α
<=> 1600/81 = tan2α
<=> tanα
= +- 40/9
=> tanα = -40/9
(Vì π ; 3 π /2)
tan(α + π/4)
<=> tan α.tan
π/4 /
1 – tanα.tan π/4
<=> (-40/9).tan π/4 / 1 – (-40/9).tan π/4
<=> -40/9.1 / 1 + 40/9.1
=> -40/49
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