Find the exact value of cos⁡(15∘)sin⁡(105∘)

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Esther Phillips · Accepted Answer

Step 1  Obtain the value of cos⁡(15∘) as follows.  Step 2  cos⁡(15∘)=cos⁡(30∘2)  =1+cos⁡(30∘)2[∴cos⁡(x2)=1+cos⁡(x)2]  =1+322  =2+34  =122+3  ≈0.95692  ≈0.95692  Step 3  Obtain the value of sin⁡(105∘) as follows.  sin⁡(105∘)=sin⁡(90∘+15∘)  =sin⁡90∘cos⁡15∘+cos⁡90∘sin⁡15∘  =(1⋅cos⁡15∘)+(0⋅sin⁡15∘)  =122+3  ≈0.95692

esfloravaou · Accepted Answer

(15∘)=cos⁡(45−30)  =cos⁡(45)cos⁡(30)+sin⁡(45)sin⁡(30)  =22×32+22×12  =24(3+1)   sin⁡(105)=sin⁡(15+90)=cos⁡15.   First find (cos⁡15). Call cos⁡15=cos⁡x   Apply the trig identity: cos⁡2x=2cos2x−1  cos⁡2x=cos⁡(30)=32=2cos2x−1  2cos2x=1+32=(2+3)2  cos2x=2+34  cos⁡x=cos⁡15=2+32. (since cos 15 is positive)  sin⁡(105)=cos⁡(15)=2+32.  Check by calculator.  sin⁡(105)=cos⁡15=0.97  2+32=1.932=0.97.

nick1337 · Accepted Answer

cos⁡(15∘) cos⁡(45−30) cos⁡(45−(30)) cos⁡(45)cos⁡(30)+sin⁡(45)sin⁡(30) The exact value of cos⁡(45) is 12 12cos⁡(30)+sin⁡(45)sin⁡(30) The exact value of cos⁡(30) is 32 12⋅32+sin⁡(45)sin⁡(30) The exact value of sin⁡(45) is 12 12⋅32+12sin⁡(30) The exact value of sin⁡(30) is 12 12⋅32+12⋅12 SImplify 12⋅32+12⋅12 3+122 The result can be shown in multiple forms. Exact form: 3+122 Decimal Form: 0.96592582.... sin⁡(105) sin⁡(45+60) sin⁡(45)cos⁡(60)+cos⁡(45)sin⁡(60) sin⁡(45)=22 22cos⁡(60)+cos⁡(45)sin⁡(60) cos⁡(60)=12 22⋅12+cos⁡(45)sin⁡(60)  cos⁡(45)=22 x22⋅12+22sin⁡(60) sin⁡(60)=32 22⋅12+22⋅32 Simplify 22⋅12+22⋅32 2+64 0.96592582...

Find the exact value of \cos(15^{\circ}) \sin(105^{\circ})

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