两道三角函数

题目

两道三角函数
1.在区间[0,pi]中,三角方程cos7x=cos5x的解的个数是?
A.5 B.7 C.11 D.13
2.COS20度/[cos35度*根号下(1-sin20度)]=___________

答案

1.
cos7x=cos5x
=>cos7x-cos5x=0
=>cos7x-cos5x=-2sin6xsinx=0
=>sin6xsinx=0.
=>sin6x=0或sinx=0
=>(1)当sin6x=0时,x=0或x=π/6或x=π/3或x=π/2或x=2π/3或x=5π/6或x=π.
(2)当sinx=0时,x=0或x=π.
综合以上两者,可得有解7个,所以,答案B.
2.
COS20度/[cos35度*根号下(1-sin20度)]
=cos20°/[cos35°*√(sin^210°-2sin10°cos10°+cos^210°)
=cos20°/[cos35°*(cos10°-sin10°)]
=cos20°/[cos35°*√2*(√2cos10°/2-√2sin10°/2)].
=cos20°/[√2cos35°*cos(10°+45°)]
=cos20°/(√2cos35°*cos55°)
=cos20°/(√2/2*(2sin35cos35°)
=√2*cos20°/sin70°
=√2*sin70°/sin70°
=√2.