三角函數(shù)測試題及答案

三角函數(shù)測試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.已知\(\sin\alpha=\frac{1}{2}\)，且\(\alpha\in(0,\frac{\pi}{2})\)，則\(\alpha=\)（）A.\(\frac{\pi}{6}\)B.\(\frac{\pi}{3}\)C.\(\frac{2\pi}{3}\)D.\(\frac{5\pi}{6}\)答案：A2.\(\cos\frac{\pi}{3}=\)（）A.\(\frac{1}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(-\frac{1}{2}\)D.\(-\frac{\sqrt{3}}{2}\)答案：A3.\(\tan\frac{\pi}{4}=\)（）A.0B.1C.-1D.\(\sqrt{3}\)答案：B4.函數(shù)\(y=\sin(x+\frac{\pi}{2})\)的圖象關(guān)于（）對稱。A.x軸B.y軸C.原點(diǎn)D.直線\(y=x\)答案：B5.在\(\triangleABC\)中，\(a=3\)，\(b=4\)，\(C=60^{\circ}\)，則\(c=\)（）A.\(\sqrt{13}\)B.\(\sqrt{37}\)C.\(\sqrt{21}\)D.\(\sqrt{19}\)答案：A6.函數(shù)\(y=\sin2x\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(4\pi\)答案：A7.\(\sin(-\frac{\pi}{6})=\)（）A.\(\frac{1}{2}\)B.\(-\frac{1}{2}\)C.\(\frac{\sqrt{3}}{2}\)D.\(-\frac{\sqrt{3}}{2}\)答案：B8.若\(\tan\alpha=3\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}=\)（）A.2B.-2C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：A9.函數(shù)\(y=\cosx\)在\([0,\pi]\)上的最大值為（）A.0B.1C.-1D.\(\frac{1}{2}\)答案：B10.已知\(\sin\alpha=\frac{3}{5}\)，\(\cos\beta=\frac{4}{5}\)，且\(\alpha,\beta\in(0,\frac{\pi}{2})\)，則\(\sin(\alpha+\beta)=\)（）A.\(\frac{24}{25}\)B.\(\frac{7}{25}\)C.\(\frac{1}{5}\)D.\(\frac{3}{5}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列與\(\sin\alpha\)相等的是（）A.\(\cos(\alpha-\frac{\pi}{2})\)B.\(\cos(\frac{\pi}{2}+\alpha)\)C.\(\sin(\pi-\alpha)\)D.\(\sin(2\pi+\alpha)\)答案：ACD2.以下函數(shù)是偶函數(shù)的是（）A.\(y=\sinx\)B.\(y=\cosx\)C.\(y=\tanx\)D.\(y=\sin2x\)答案：B3.對于\(y=A\sin(\omegax+\varphi)\)（\(A>0,\omega>0\)），其振幅為（）A.\(A\)B.\(\omega\)C.\(\varphi\)D.\(\frac{2\pi}{\omega}\)答案：A4.在\(\triangleABC\)中，由正弦定理可得（）A.\(\frac{a}{\sinA}=\frac{\sinB}\)B.\(\frac{a}{\sinB}=\frac{\sinA}\)C.\(\frac{a}{\cosA}=\frac{\cosB}\)D.\(\frac{a}{\cosB}=\frac{\cosA}\)答案：A5.當(dāng)\(\alpha\in(0,\pi)\)時(shí)，\(\sin\alpha+\cos\alpha>0\)的區(qū)間是（）A.\((0,\frac{\pi}{2})\)B.\((\frac{\pi}{4},\pi)\)C.\((\frac{\pi}{4},\frac{3\pi}{4})\)D.\((0,\frac{3\pi}{4})\)答案：D6.函數(shù)\(y=\tanx\)的周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(-\pi\)答案：A7.下列等式成立的是（）A.\(\sin^{2}\alpha+\cos^{2}\alpha=1\)B.\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\)C.\(\sin2\alpha=2\sin\alpha\cos\alpha\)D.\(\cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha\)答案：ABCD8.若\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\pi)\)，則\(\sin\alpha=\)（）A.\(\frac{2\sqrt{2}}{3}\)B.\(-\frac{2\sqrt{2}}{3}\)C.\(\frac{\sqrt{10}}{3}\)D.\(-\frac{\sqrt{10}}{3}\)答案：A9.函數(shù)\(y=\cos(x-\frac{\pi}{3})\)的圖象是由\(y=\cosx\)的圖象（）得到的。A.向右平移\(\frac{\pi}{3}\)個單位B.向左平移\(\frac{\pi}{3}\)個單位C.向上平移\(\frac{\pi}{3}\)個單位D.向下平移\(\frac{\pi}{3}\)個單位答案：A10.在\(\triangleABC\)中，若\(a=5\)，\(b=8\)，\(A=30^{\circ}\)，則三角形解的個數(shù)為（）A.0B.1C.2D.3答案：C三、判斷題（每題2分，共10題）1.\(\sin150^{\circ}=\frac{1}{2}\)。（）答案：對2.函數(shù)\(y=\sinx\)在\((-\frac{\pi}{2},\frac{\pi}{2})\)上是單調(diào)遞減的。（）答案：錯3.\(\cos(-\alpha)=-\cos\alpha\)。（）答案：錯4.對于\(\triangleABC\)，\(a^{2}=b^{2}+c^{2}-2bc\cosA\)。（）答案：對5.\(\tan210^{\circ}=\frac{\sqrt{3}}{3}\)。（）答案：對6.函數(shù)\(y=\cos2x\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{4},0)\)對稱。（）答案：錯7.若\(\sin\alpha=\sin\beta\)，則\(\alpha=\beta\)。（）答案：錯8.函數(shù)\(y=\sinx+\cosx\)的最大值是\(\sqrt{2}\)。（）答案：對9.在\(\triangleABC\)中，\(\sinA>\sinB\)等價(jià)于\(A>B\)。（）答案：對10.\(\cos\frac{7\pi}{6}=-\frac{\sqrt{3}}{2}\)。（）答案：對四、簡答題（每題5分，共4題）1.求\(\sin75^{\circ}\)的值。答案：\(\sin75^{\circ}=\sin(45^{\circ}+30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\)2.簡述函數(shù)\(y=A\sin(\omegax+\varphi)\)（\(A>0,\omega>0\)）的圖象變換規(guī)律。答案：\(A\)影響振幅，\(\omega\)影響周期\(T=\frac{2\pi}{\omega}\)，\(\varphi\)影響左右平移，左加右減。3.在\(\triangleABC\)中，已知\(a=2\)，\(b=3\)，\(C=120^{\circ}\)，求\(c\)的值。答案：根據(jù)余弦定理\(c^{2}=a^{2}+b^{2}-2ab\cosC\)，\(c^{2}=4+9-2\times2\times3\times(-\frac{1}{2})=19\)，\(c=\sqrt{19}\)。4.求\(y=\sinx\)在\([0,2\pi]\)上的單調(diào)遞增區(qū)間。答案：\([0,\frac{\pi}{2}]\)和\([\frac{3\pi}{2},2\pi]\)。五、討論題（每題5分，共4題）1.討論\(\sin\alpha\)與\(\cos\alpha\)在\([0,2\pi]\)上的大小關(guān)系。答案：在\([0,\frac{\pi}{4})\)和\((\frac{5\pi}{4},2\pi]\)上\(\cos\alpha>\sin\alpha\)，在\((\frac{\pi}{4},\frac{5\pi}{4})\)上\(\sin\alpha>\cos\alpha\)。2.討論如何根據(jù)\(a,b,A\)（\(a,b\)為三角形的兩邊，\(A\)為\(a,b\)夾角）判斷三角形的形狀。答案：根據(jù)余弦定理\(a^{2}=b^{2}+c^{2}-2bc\cosA\)，若\(a^{2}=b^{2}+c^{2}\)為直角三角形，\(a^{2}<b^{2}+c^{2}\)為銳角三角形，\(a^{2}>b^{2}+c^{2}\)為鈍角三角形。3.討論函數(shù)\(y=\sinx\)與\(y=\cosx\)圖象的聯(lián)系。答案：\