题目
Problem
(a) Express 4 cos x − sin x 4\cos x-\sin x 4 cos x − sin x R cos ( x + α ) R\cos(x+\alpha) R cos ( x + α ) R > 0 R>0 R > 0 0 ∘ < α < 90 ∘ 0^\circ<\alpha<90^\circ 0 ∘ < α < 9 0 ∘ R R R α \alpha α 2 2 2
(3)
(b) Hence solve the equation 4 cos 2 x − sin 2 x = 3 4\cos2x-\sin2x=3 4 cos 2 x − sin 2 x = 3 0 ∘ < x < 180 ∘ 0^\circ<x<180^\circ 0 ∘ < x < 18 0 ∘
(5)
题目中文翻译
(a) 将 4 cos x − sin x 4\cos x-\sin x 4 cos x − sin x R cos ( x + α ) R\cos(x+\alpha) R cos ( x + α ) R > 0 R>0 R > 0 0 ∘ < α < 90 ∘ 0^\circ<\alpha<90^\circ 0 ∘ < α < 9 0 ∘ R R R α \alpha α 2 2 2
(3)
(b) 由此解方程 4 cos 2 x − sin 2 x = 3 4\cos2x-\sin2x=3 4 cos 2 x − sin 2 x = 3 0 ∘ < x < 180 ∘ 0^\circ<x<180^\circ 0 ∘ < x < 18 0 ∘
(5)
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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展开 R cos ( x + α ) R\cos(x+\alpha) R cos ( x + α ) α \alpha α
答题过程
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Expanding,
R cos ( x + α ) = R cos α cos x − R sin α sin x . R\cos(x+\alpha)=R\cos\alpha\cos x-R\sin\alpha\sin x. R cos ( x + α ) = R cos α cos x − R sin α sin x . Thus
R cos α = 4 , R sin α = 1. R\cos\alpha=4,\qquad R\sin\alpha=1. R cos α = 4 , R sin α = 1. Therefore
R = 4 2 + 1 2 = 17 , α = tan − 1 1 4 = 14.036243 … ∘ . \begin{align*}
R=&\,\sqrt{4^2+1^2}=\sqrt{17},\\[4mm]
\alpha=&\,\tan^{-1}\frac14\\[4mm]
=&\,14.036243\ldots^\circ.
\end{align*} R = α = = 4 2 + 1 2 = 17 , tan − 1 4 1 14.036243 … ∘ . Hence R = 17 , α = 14.04 ∘ \boxed{R=\sqrt{17},\quad\alpha=14.04^\circ} R = 17 , α = 14.0 4 ∘
(b)
解法一
思路
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承接(a)，先求 2 x + α 2x+\alpha 2 x + α 2 2 2
答题过程
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Using part (a), with the unrounded value of α \alpha α
17 cos ( 2 x + α ) = 3. \sqrt{17}\cos(2x+\alpha)=3. 17 cos ( 2 x + α ) = 3. Let
β = cos − 1 3 17 = 43.313856 … ∘ . \beta=\cos^{-1}\frac3{\sqrt{17}}=43.313856\ldots^\circ. β = cos − 1 17 3 = 43.313856 … ∘ . The values giving 0 ∘ < 2 x < 360 ∘ 0^\circ<2x<360^\circ 0 ∘ < 2 x < 36 0 ∘
2 x = β − α or 2 x = 360 ∘ − β − α . \begin{align*}
2x=&\,\beta-\alpha\\[4mm]
\text{or}\quad 2x=&\,360^\circ-\beta-\alpha.
\end{align*} 2 x = or 2 x = β − α 36 0 ∘ − β − α . Thus
x = 14.6 ∘ or 151.3 ∘ . \boxed{x=14.6^\circ\quad\text{or}\quad151.3^\circ}. x = 14. 6 ∘ or 151. 3 ∘ . 