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【330957】4.1 正弦与余弦

时间:2025-01-21 13:27:20 作者: 字数:1884字
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第四章 锐角三角函数

4.1 正弦与余弦


基础导练

1.Rt△ABC中,∠C=90°BC=2sin A= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> ,则边AC的长是( )

A. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> B.3 C. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> D. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>

2.已知△ABC中,AC=4BC=3AB=5,则sin A=( )

A. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> B. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> C. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> D. <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>

3.在Rt△ABC中,∠C=90°sin A= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> ,则cos B的值等于(

A <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> B <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> C <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> D <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>

4.如图在 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> 中, <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> ,则 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> = =  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> = = .

5.如图,在Rt△ABC中,∠C90°,我们把锐角∠A的邻边b与斜边c的比叫做∠A______,记作=_________,即cos A= =_____

6.在Rt△ABC中,∠C=90°AB=5BC=4,则cos A =

7.如图,在Rt△ABC中,∠C=900BC=6sin A= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> ,求cos Atan B的值.

 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>


能力提升

8.如图,在四边形ABCD中,AD∥BCAC⊥ABAD=CDcos B= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> BC=26.

(1)cos∠DAC的值;

(2)求线段AD的长.

 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>


9.在矩形ABCD中,DC=23CF⊥BD分别交BDAD于点EF,连接BF.

(1)求证:△DEC∽△FDC

2)当FAD的中点时,求sin∠FBD的值及BC的长度.

 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>


参考答案

1A 2A 3B

4 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> 5余弦  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> 6.  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>

7解: <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .  <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a>

8解:(1)Rt△ABC中,∠BAC=90°cos B= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .

因为BC=26所以AB=10.所以AC=24.

因为AD∥BC所以DAC=∠ACB.所以cos∠DAC=cos∠ACB= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .

(2)过点DDE⊥AC,垂足为E.

因为AD=DC所以AE=EC=12.所以Rt△ADE中,cos∠DAE= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .所以AD=13.

9.(1)证明:因为矩形ABCD,CF⊥BD,∠DEC=∠FDC=90°,又∠DCE=∠FCD所以DEC∽△FDC.

(2)解:因为FAD的中点,AD∥BC所以FB=FC所以sin∠FBD= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .EF=x,则FC=3x,CE=2x.因为DEC∽△FDC,得x=2.所以CF= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .Rt△CFD中,DF= <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> 所以BC=2DF=2 <a href="/tags/942/" title="正弦" class="c1" target="_blank">正弦</a> <a href="/tags/943/" title="余弦" class="c1" target="_blank">余弦</a> .