Teddy asked in Science & MathematicsMathematics · 2 months ago

# Sorry, the last question my teacher just put this in the homework and I don't know what he wants me to do like how to solve it? ?

cos 82.5 degrees

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• 2 months ago

cos 82.5 degrees = 0.130526...

• Philip
Lv 6
2 months ago

Put a = 82.5. Then 2a = 165 = 120 + 45. cos(2a) = cos(120)cos(45) - sin(120)*

sin(45) = -cos(60)cos(45) - sin(60)sin(45) = -((1/2)(1/2)rt2 + (1/2)rt3(1/2)rt2) =

-(1/4)rt2(1+rt3) where rt = "square root of".;

Now cos(2a) = 1-2sin^2(a) = -(1/4)rt2(1+rt3), ie., 2sin^2(a) = 1+ (1/4)rt2(1+rt3)

= (1/4)(4+rt2(1+rt3)). Then sin^2(a) = (1/8)(4+rt2+rt6);

cos^2(a) = 1 - sin^2(a) = (8-(4+rt2+rt6))/8 = (4-rt2-rt6)/8 = 0.01703708686 and

cos(a) = 0.1305261922. Check with calculator gives arccos(0.1305261922) =

82.5 degrees. The exact value of cos(a) is rt((4-rt2-rt6)/8).

• Alan
Lv 7
2 months ago

using a calculator is all that is required.

If you are expected to find an exact answer.

cos(82.5) = cos(90- 7.5) = cos(90)cos(7.5) + sin(90) sin(7.5)

cos(82.5) = sin(7.5)

sin(30) = 1/2

cos(30) = sqrt(3)/2

sin (x/2) = +/- sqrt( (1-cos(x)) /2 )

sin(15) = +/- sqrt ( (1 - sqrt(3)/2 )/2 ) =

sin(15 ) = sqrt ( (1/2) - sqrt(3)/4) = sqrt ( 2/4 - sqrt(3)/4 ) =

sin(15) = (1/2) sqrt( 2 - sqrt(3))

cos(15) = sqrt( 1 - ((1/2) sqrt(2 -sqrt(3))^2 )

cos(15 ) = sqrt( 1 - (1/4) (2-sqrt(3) )

cos(15) = sqrt( 1 -1/2 + sqrt(3)/4 )

cos(15 ) = sqrt( 2/4 + sqrt(3)/4 ) = (1/2) sqrt( 2 + sqrt(3) )

sin(7.5) = + sqrt( (1 - (1/2)*sqrt(2 + sqrt(3) ) /2 )

sin(7.5) = + sqrt( (1/2) - (1/4)*sqrt(2+sqrt(3)) )

sin(7.5) = + sqrt( (2/4) - (1/4) sqrt(2+sqrt(3))

sin(7.5) = + (1/2) sqrt( 2 -sqrt(2+ sqrt(3)) )

cos(82.5) = + (1/2) sqrt( 2 -sqrt(2+ sqrt(3)) )

I'm sure they are ways to simplify this.

Maybe you could ask how to simplify this as another question.

• Bill-M
Lv 7
2 months ago

The Cosine of 82.5 degrees is 0.1305