✔ 最佳答案
1.
tan(22° + 23°) = tan45°
(tan22° + tan23°) / (1 - tan22°tan23°) = 1
tan22° + tan23° = 1 - tan22°tan23°
tan22°tan23° + tan22° + tan23° = 1
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2.
sin159°sin51° + sin69°sin39°
= sin(180°-21°)sin(90°-39°) + sin(90°-21°)sin39°
= sin21°cos39° + cos21°sin39°
= sin(21°+39°)
= sin60°
= (√3)/2
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3.
sinθ + cosθ = 1/3
(sinθ + cosθ)² = (1/3)²
(sin²θ + cos²θ) + 2sinθcosθ = 1/9
1 + 2sinθcosθ = 1/9
1 + sin2θ = 1/9
sin2θ = -8/9
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4.
270° < θ < 360°
sinθ < 0 及 cosθ > 0
cosθ = 1/3
sinθ
= -√[1 - cos²θ]
= -√[1 - (1/3)²]
= -√(8/9)
= -2(√2)/3
sin2θ
= 2sinθcosθ
= 2 x [(-2√2)/3] x (1/3)
= -4(√2)/9
cos2θ
= cos²θ - sin²θ
= (1/3)² - (-√8/9)²
= -7/9
tan2θ
= sin2θ / cos2θ
= [-4(√2)/9] / (-7/9)
= 4(√2)/7
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5.
90° < θ < 180°
sinθ > 0 及 cosθ < 0
45° < θ/2 < 90°
sin(θ/2) > 0 及 cos(θ/2) > 0
sinθ = 4/5
cosθ
= -√[1 - sin²θ]
= -√[1 - (4/5)²]
= -√(9/25)
= -3/5
2cos²(θ/2) - 1 = cosθ
2cos²(θ/2) - 1 = -3/5
2cos²(θ/2) = 2/5
cos²(θ/2) = 1/5
由於 cos(θ/2) >0
故此 cos(θ/2) = √(1/5)= (√5)/5
1 - 2sin²(θ/2) = cosθ
1 - 2sin²(θ/2) = -3/5
2sin²(θ/2) = 8/5
sin²(θ/2) = 4/5
由於 sin(θ/2)> 0
故此 sin(θ/2) = √(4/5) = 2(√5)/5
tan(θ/2)
= sin(θ/2) / cos(θ/2)= [2(√5)/5] / [(√5)/5]
= 2
