**Isosceles triangle has its head A Where the angle A**

ABCABC

**B**C = 2*π/5*

*and k is a relative integer , we put BC=a . *

*Bisector of the angle ABC cut the line segment [AC] In point D.*

**[DH],[DI],[AJ] and [BK] **Are the elevations in the triangles **DAB , ****DBC , ABC , DBC ***Respectively.*

*1. **complete the shape.*

*2.*

* a. **Prove that the two triangles DAB and DBC are Isosceles.*

* b.Consider the triangle, Express length AB a function of cos(*

**π/5)**and**a ,***Infer the phrase*

**CD***a function of*

**cos(**

**π/5)**and**a*** c. **Express the **triangle DBC , *

*Proved that,*

*CD*

**=**

*2a*

**cos(2**

**π/5)*** d. **Infer that ***cos(****π/5)-****cos(2***π/5)*=*1/2*

*3. *

* a. **Express the **triangle DBC , *

*Express length*

**IB**a function of**cos(**

**π/5)**and**a ,**

*Infer the phrase*

**CD***a function of*

**cos(**

**π/5)**and**a** b. *Prove that **IC*=* 2a [cos²(***2****π/5)]**

** c. ***Infer that ***cos(****π/5)+***cos²(***2****π/5)=1**

**4.**

** **a. *Solve in ℝ *

*the following equation:*

**x²+1/2 x-1/4**** **b. *Check that *

**cos(**

**π/5)**

**is a solution to this equation*** c. **Prove that for every real number x it is: cos(sinx)>sin(cosx)*

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