题目
Problem
(a) Express 7 sin θ + 24 cos θ 7\sin\theta+24\cos\theta 7 sin θ + 24 cos θ R cos ( θ − α ) R\cos(\theta-\alpha) R cos ( θ − α ) R > 0 R>0 R > 0 0 < α < 1 2 π 0<\alpha<\frac12\pi 0 < α < 2 1 π α \alpha α 4 4 4
(3)
(b) Hence solve the equation 7 sin 1 3 x + 24 cos 1 3 x = 24.5 7\sin\frac13x+24\cos\frac13x=24.5 7 sin 3 1 x + 24 cos 3 1 x = 24.5 0 < x < π 0<x<\pi 0 < x < π
(4)
题目中文翻译
(a) 将 7 sin θ + 24 cos θ 7\sin\theta+24\cos\theta 7 sin θ + 24 cos θ R cos ( θ − α ) R\cos(\theta-\alpha) R cos ( θ − α ) R > 0 R>0 R > 0 0 < α < 1 2 π 0<\alpha<\frac12\pi 0 < α < 2 1 π α \alpha α 4 4 4
(3)
(b) 由此解方程 7 sin 1 3 x + 24 cos 1 3 x = 24.5 7\sin\frac13x+24\cos\frac13x=24.5 7 sin 3 1 x + 24 cos 3 1 x = 24.5 0 < x < π 0<x<\pi 0 < x < π
(4)
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
(a)
解法一
思路
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与余弦差角展开比较系数，振幅由平方和求出，相位取第一象限。
答题过程
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Since
R cos ( θ − α ) = R cos α cos θ + R sin α sin θ , R\cos(\theta-\alpha)
=R\cos\alpha\cos\theta+R\sin\alpha\sin\theta, R cos ( θ − α ) = R cos α cos θ + R sin α sin θ , we require
R cos α = 24 , R sin α = 7. \begin{align*}
R\cos\alpha=&\,24,\\[4mm]
R\sin\alpha=&\,7.
\end{align*} R cos α = R sin α = 24 , 7. Hence
R = 24 2 + 7 2 = 25 , R=\sqrt{24^2+7^2}=25, R = 2 4 2 + 7 2 = 25 , and
α = tan − 1 7 24 = 0.283794 … = 0.2838 . \begin{align*}
\alpha=&\,\tan^{-1}\frac7{24}\\[4mm]
=&\,0.283794\ldots=\boxed{0.2838}.
\end{align*} α = = tan − 1 24 7 0.283794 … = 0.2838 . Thus the required form is 25 cos ( θ − α ) \boxed{25\cos(\theta-\alpha)} 25 cos ( θ − α )
(b)
解法一
思路
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把(a)中的角换成 x / 3 x/3 x /3
答题过程
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Using (a),
cos ( x / 3 − α ) = 24.5 25 = 0.98 , \cos(x/3-\alpha)=\frac{24.5}{25}=0.98, cos ( x /3 − α ) = 25 24.5 = 0.98 , where α = tan − 1 ( 7 / 24 ) \alpha=\tan^{-1}(7/24) α = tan − 1 ( 7/24 ) β = cos − 1 ( 0.98 ) \beta=\cos^{-1}(0.98) β = cos − 1 ( 0.98 )
Since 0 < x < π 0<x<\pi 0 < x < π
− α < x / 3 − α < π / 3 − α . -\alpha<x/3-\alpha<\pi/3-\alpha. − α < x /3 − α < π /3 − α . Both β \beta β − β -\beta − β
x = 3 ( α − β ) = 0.250377 … x=3(\alpha-\beta)=0.250377\ldots x = 3 ( α − β ) = 0.250377 … or
x = 3 ( α + β ) = 1.452386 … . x=3(\alpha+\beta)=1.452386\ldots. x = 3 ( α + β ) = 1.452386 … . Therefore
x = 0.250 , 1.45 radians . \boxed{x=0.250,\ 1.45\text{ radians}}. x = 0.250 , 1.45 radians . 