maths...20pts!

chi man

2009-10-18 8:27 am

The angle α is in the 3rd quadrant and satisfies cos^2 α = 9/25, β is in 2nd qurdrant and satisfies sinβ = 12/13, while γ is in the 4th quadrant and satisfies tanγ = 7/24. Find the precise value of each of the following quantities, explaining every step of ur argument.

a) cos(α+β)
b) sin(α+γ)
c) tan(α-β)
d) sin(β+γ)

回答 (1)

用戶9112

2009-10-18 8:59 am

✔ 最佳答案

First quadrant sin & cos +ve
Second quadrant sin +ve, cos -ve
Third quadrant sin & cos - ve
Fourth quadrant sin -ve, cos +ve
Third quadrant : cos^2α = 9/25
cosα = -3/5, sinα = -4/5, tanα = 4/3
Second quadrant sinβ = 12/13
cosβ = -5/13, tanβ = -12/5
Fourth quadrant : tanγ = -7/24 (cannot be positive)
sinγ = -7/25, cosγ = 24/25
(a) cos(α + β) = cosα cosβ - sinα sinβ
= (-3/5)(-5/13) - (-4/5)(12/13)
= 3/13 + 48/65
= 63/65
(b) sin(α + γ) = sinα cosγ + sinγ cosα
= (-4/5)(24/25) + (-7/25)(-3/5)
= -96/125 + 21/125
= -75/125
= -3/5
(c) tan(α - β) = (tanα - tanβ) / (1 + tanα tanβ)
= [(4/3) - (-12/5)] / [1 + (4/3)(-12/5)]
= [(20 + 36)/15] / [1 - 48/15]
= -56/33
(d) sin(β + γ) = sinβ cosγ + sinγ cosβ
= (12/13)(24/25) + (-7/25)(-5/13)
= 288/325 + 35/325
= 323/325

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