全國(guó)中學(xué)生數(shù)學(xué)競(jìng)賽三角函數(shù)題試題及答案1.已知\(\sin\alpha=\frac{3}{5}\)，且\(\alpha\)為第二象限角，則\(\cos\alpha\)的值為（）A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)答案：B2.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(4\pi\)答案：A3.若\(\tan\alpha=3\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值為（）A.2B.-2C.1D.-1答案：A4.已知\(\cos15^{\circ}\)的值為（）A.\(\frac{\sqrt{6}-\sqrt{2}}{4}\)B.\(\frac{\sqrt{6}+\sqrt{2}}{4}\)C.\(\frac{\sqrt{3}-1}{2\sqrt{2}}\)D.\(\frac{\sqrt{3}+1}{2\sqrt{2}}\)答案：B5.函數(shù)\(y=\cos^2x-\sin^2x\)的化簡(jiǎn)結(jié)果是（）A.\(\cos2x\)B.\(\sin2x\)C.\(-\cos2x\)D.\(-\sin2x\)答案：A6.若\(\sin\theta=\frac{m-3}{m+5}\)，\(\cos\theta=\frac{4-2m}{m+5}\)，且\(\theta\)為第二象限角，則\(m\)的取值范圍是（）A.\(m=8\)B.\(m=0\)C.\(m=8\)或\(m=0\)D.\(3<m<9\)答案：A7.已知\(\sin(\alpha+\beta)=\frac{1}{2}\)，\(\sin(\alpha-\beta)=\frac{1}{3}\)則\(\frac{\tan\alpha}{\tan\beta}\)的值為（）A.5B.-1C.6D.\(\frac{1}{6}\)答案：A8.函數(shù)\(y=A\sin(\omegax+\varphi)\)（\(A>0\)，\(\omega>0\)）的圖象的一個(gè)對(duì)稱中心為\((\frac{\pi}{12},0)\)，則\(\omega\)的值可以是（）A.2B.3C.4D.5答案：B9.已知\(\cos(\frac{\pi}{4}-\alpha)=\frac{3}{5}\)，則\(\sin2\alpha\)的值為（）A.\(\frac{19}{25}\)B.\(\frac{16}{25}\)C.\(\frac{14}{25}\)D.\(\frac{7}{25}\)答案：A10.若\(\sin\theta+\cos\theta=\frac{1}{5}\)，\(\theta\in(0,\pi)\)，則\(\tan\theta\)的值為（）A.\(-\frac{3}{4}\)B.\(-\frac{4}{3}\)C.\(\frac{3}{4}\)D.\(\frac{4}{3}\)答案：B11.函數(shù)\(y=\sinx\cosx+\sinx+\cosx\)的最大值是（）A.\(\frac{1}{2}+\sqrt{2}\)B.\(\frac{1}{2}-\sqrt{2}\)C.\(1+\sqrt{2}\)D.\(1-\sqrt{2}\)答案：A12.已知\(\sin\alpha=\frac{1}{3}\)，則\(\sin(\alpha+\frac{\pi}{2})\sin(\alpha-\frac{3\pi}{2})\)的值為（）A.\(\frac{8}{9}\)B.\(-\frac{8}{9}\)C.\(\frac{1}{9}\)D.\(-\frac{1}{9}\)答案：A13.若\(\tan\frac{\alpha}{2}=2\)，則\(\frac{1-\cos\alpha+\sin\alpha}{1+\cos\alpha+\sin\alpha}\)的值為（）A.2B.-2C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：A14.函數(shù)\(y=\sin(x+\frac{\pi}{6})\cos(x+\frac{\pi}{6})\)的最小正周期是（）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)答案：B15.已知\(\sin\alpha\cos\alpha=\frac{1}{8}\)，且\(\frac{\pi}{4}<\alpha<\frac{\pi}{2}\)，則\(\cos\alpha-\sin\alpha\)的值為（）A.\(\frac{\sqrt{3}}{2}\)B.\(-\frac{\sqrt{3}}{2}\)C.\(\frac{\sqrt{3}}{4}\)D.\(-\frac{\sqrt{3}}{4}\)答案：B16.函數(shù)\(y=\frac{\sinx}{1+\cosx}\)的最小正周期是（）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)答案：B17.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})\)的值為（）A.\(\frac{1}{7}\)B.7C.\(-\frac{1}{7}\)D.-7答案：A18.若\(\cos\alpha=-\frac{4}{5}\)，\(\alpha\)是第三象限角，則\(\sin(\alpha+\frac{\pi}{4})\)的值為（）A.\(-\frac{7\sqrt{2}}{10}\)B.\(\frac{7\sqrt{2}}{10}\)C.\(-\frac{\sqrt{2}}{10}\)D.\(\frac{\sqrt{2}}{10}\)答案：A19.函數(shù)\(y=\sin^4x+\cos^4x\)的最小正周期是（）A.\(\frac{\pi}{4}\)B.\(\frac{\pi}{2}\)C.\(\pi\)D.\(2\pi\)答案：B20.已知\(\sin\alpha+\cos\alpha=\frac{1}{5}\)，則\(\sin^3\alpha+\cos^3\alpha\)的值為（）A.\(\frac{37}{125}\)B.\(\frac{11}{125}\)C.\(\frac{19}{125}\)D.\(\frac{22}{125}\)答案：A1.以下哪些函數(shù)是三角函數(shù)（）A.\(y=\sinx\)B.\(y=\cosx\)C.\(y=\tanx\)D.\(y=x^2\)答案：ABC2.三角函數(shù)的基本性質(zhì)包括（）A.周期性B.奇偶性C.單調(diào)性D.對(duì)稱性答案：ABCD3.下列等式成立的是（）A.\(\sin^2x+\cos^2x=1\)B.\(\tanx=\frac{\sinx}{\cosx}\)C.\(\sin(A+B)=\sinA\cosB+\cosA\sinB\)D.\(\cos(A-B)=\cosA\cosB-\sinA\sinB\)答案：ABC4.函數(shù)\(y=A\sin(\omegax+\varphi)\)（\(A\neq0\)，\(\omega>0\)）中，\(A\)的作用是（）A.決定函數(shù)的振幅B.影響函數(shù)的周期C.決定函數(shù)的最值D.不影響函數(shù)的形狀答案：AC5.關(guān)于三角函數(shù)的圖象，以下說(shuō)法正確的是（）A.\(y=\sinx\)的圖象關(guān)于原點(diǎn)對(duì)稱B.\(y=\cosx\)的圖象關(guān)于\(y\)軸對(duì)稱C.\(y=\tanx\)的圖象有無(wú)數(shù)條漸近線D.\(y=\sinx\)與\(y=\cosx\)的圖象形狀相同答案：ABC6.已知\(\sin\alpha=\frac{1}{2}\)，則\(\alpha\)可能的值為（）A.\(\frac{\pi}{6}\)B.\(\frac{5\pi}{6}\)C.\(\frac{7\pi}{6}\)D.\(\frac{11\pi}{6}\)答案：ABCD7.若\(\cos\theta=-\frac{3}{5}\)，且\(\theta\)為第二象限角，則\(\sin\theta\)的值為（）A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)答案：A8.函數(shù)\(y=\sinx\)在區(qū)間\([0,2\pi]\)上的單調(diào)遞增區(qū)間是（）A.\([0,\frac{\pi}{2}]\)B.\([\frac{\pi}{2},\frac{3\pi}{2}]\)C.\([\frac{3\pi}{2},2\pi]\)D.以上都不對(duì)答案：AC9.以下哪些是誘導(dǎo)公式（）A.\(\sin(\pi-\alpha)=\sin\alpha\)B.\(\cos(\pi+\alpha)=-\cos\alpha\)C.\(\tan(-\alpha)=-\tan\alpha\)D.\(\sin(\frac{\pi}{2}-\alpha)=\cos\alpha\)答案：ABCD10.已知\(\tan\alpha=2\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值為（）A.3B.\(\frac{1}{3}\)C.-3D.\(-\frac{1}{3}\)答案：A1.三角函數(shù)\(y=\sinx\)的最大值是1。（）答案：√2.\(\cos^2x-\sin^2x=\cos2x\)恒成立。（）答案：√3.函數(shù)\(y=\tanx\)的定義域是\(x\neqk\pi+\frac{\pi}{2},k\inZ\)。（）答案：√4.\(\sin(\alpha+\beta)=\sin\alpha+\sin\beta\)。（）答案：×5.\(\cos(-\alpha)=-\cos\alpha\)。（）答案：×6.函數(shù)\(y=\sinx\)是奇函數(shù)。（）答案：√7.若\(\sin\alpha=0\)，則\(\alpha=0\)。（）答案：×8.三角函數(shù)的圖象平移不改變函數(shù)的性質(zhì)。（）答案：×9.\(\tan45^{\circ}=1\)。（）答案：√10.函數(shù)\(y=A\sin(\omegax+\varphi)\)的初相是\(\varphi\)。（）答案：√1.\(\sin30^{\circ}=\)（）答案：\(\frac{1}{2}\)2.\(\cos60^{\circ}=\)（）答案：\(\frac{1}{2}\)3.\(\tan45^{\circ}=\)（）答案：14.函數(shù)\(y=\sinx\)的最小正周期\(T=\)（）答案：\(2\pi\)5.已知\(\sin\alpha=\frac{1}{2}\)，\(\alpha\in(0,\frac{\pi}{2})\)，則\(\alpha=\)（）答案：\(\frac{\pi}{6}\)6.若\(\cos\theta=-\frac{1}{2}\)，\(\theta\)為第二象限角，則\(\theta=\)（）答案：\(\frac{2\pi}{3}\)7.函數(shù)\(y=\cosx\)在\([0,\pi]\)上的最大值是（）答案：18.\(\sin(\pi-\alpha)=\)（）答案：\(\sin\alpha\)9.\(\cos(\frac{\pi}{2}-\alpha)=\)（）答案：\(\sin\alpha\)10.已知\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\)，當(dāng)\(\sin\alpha=\frac{3}{5}\)，\(\cos\alpha=