How do i simplify this?

Question

Hi my problem is the following.

 

Use the sum and difference identity to find the exact value of sine cosine and tangent.

195°=60°+135°

 

And here is what i did

Sin(195)=sin(60+135)

=sin60cos135+cos60sin135

 

But i dont know how to simplify that result

 

I did the samething with cos

Cos(195)=cos(60+135)

=cos60cos135-sin60sin135

 

And i have the same problem i dont get how to simplify that?

Can someone help please and how do it for tangent?

 

Thank you:)

Answer 1

Use the sum and difference identity to find the exact values of sine, cosine, and tangent of 195°=60°+135°.

 

30°-60°-90°     45°-45°-90°

 

      /|                   |\

2  /  |                1 |   \ √(2)

  /    | √(3)           |      \

/      |                   |        \            

    1                          -1

 

sin(195°)=

sin(60°+135°)=

sin60°cos135°+cos60°sin135°=

(√(3)/2)(-1/√(2))+(1/2)(1/√(2))=

(√(3)/2)(-√(2)/2)+(1/2)(√(2)/2)=
(-√(6)/4)+(√(2)/4)=

(√(2)-√(6))/4 = the exact value of sin(195°)

cos(195°)=

cos(60°+135°)=

cos60°cos135°-sin60°sin135°=

(1/2)(-√(2)/2)-(√(3)/2)(√(2)/2)=

(-√(2)/4)-(√(6)/4)=

-(√(2)+√(6))/4 = the exact value of cos(195°)

 

tan(195°)=

sin(195°)/cos(195°) =

-(√(2)-√(6))/(√(2)+√(6)) * (√(2)-√(6))/(√(2)-√(6))=

-(2-2√(12)+6)/(2-6)=

(8-4√(3))/4=

2-√(3) = the exact value of tan(195°)

 

check:

 

calculator values                              radical values    

sin(195°)≈ -0.25881904510252        (√(2)-√(6))/4≈ -0.25881904510252

cos(195°)≈ -0.965925826289068     -(√(2)+√(6))/4≈ -0.96592582628907

tan(195°)≈ 0.267949192431122       2-√(3)≈ 0.26794919243112

Answer 2

sin(195)=sin(60+135)

sin(60+135)=sin(60)cos(135)+cos(60)sin(135)

                   =[sqrt(3)/2](-0.70710678)+(1/2)(0.70710678)

                   =(1.7320508/2)(-0.70710678)+(1/2)(0.70710678)

                   =(0.8660254)(-0.70710678)+(1/2)(0.70710678)

                   =-0.6123724+0.3535534

                   =-0.258819

 

cos(195)=cos(60+135)

cos(60+135)=cos(60)cos(135)-sin(60)sin(135)

                    =(1/2)((-0.70710678)-(sqrt[3]/2)(0.70710678)

                    =-0.3535534-(0.8660254)(0.70710678)

                    =-0.3535534-0.6123724

                    =-0.9659258

 

tan(195)=tan(60+135)

tan(60+135)=[tan(60)+tan(135)]/(1-tan(60)tan(135))

                   =[1.7320508+(-1)]/(1-[1.7320508][-1])             (tan(60)=sqrt(3)=1.7320508)

                   =(0.7320508)/(1+1.7320508)

                   =(0.7320508)/(2.7320508)

                   =0.2679492

Answer 3

 The formula is correct.

 

 To evaluate the numeric values of Sin, and cos.

 

  Have to use calculator and figure out the values of each factor, and the value of the whose expression .

 

 

  i.e.   Sin 60 =0.866    Cos 195=-0.96

 

 You can check your result too.