1 Simplify to an expression of a single trigonometric functi

1.) Simplify to an expression of a single trigonometric function, then find the exact value.

2.) Simplify the equation to a single trigonometric function.

3.) Write the equation as a single sine expression.

4.) Find the exact value of cos 75 degrees.

Solution

1) 6 cos² (/8) - 6 sin² (/8)

==> 6 (cos² (/8) - sin² (/8))

==> 6 cos (2(/8))                since cos²x - sin²x = cos(2x) , here x = /8

==> 6 cos(/4)

==> 6(1/2)

==> 32

==> 4.2426

6 cos² (/8) - 6 sin² (/8) = 4.2426

Hence 6 cos² (/8) - 6 sin² (/8) = 4.2426

2) y = 10 sin(x/2) cos(x/2)

==> y = 5 (2sin(x/2) cos(x/2))

==> y = 5 sin(2 * x/2)            since 2 sin cos = sin(2) , here = x/2

==> y = 5sinx

Hence y = 10 sin(x/2) cos(x/2) = 5sinx

3) sin8 cos4 - cos8 sin4

==> sin(8 - 4)            since sinA cosB - cosA sinB , here A = 8 , B = 4

==> sin(4)

Hence sin8 cos4 - cos8 sin4 = sin(4)

4) cos 75^o = cos(45^o + 30^o)

we have cos(A + B) = cosAcosB - sinAsinB

==> cos(45^o + 30^o) = cos45^ocos30^o - sin45^osin30^o

==> (1/2)(3/2) - (1/2)(1/2)

==> (3 -1)/(22)

Hence cos 75^o = (3 -1)/(22)

==> cos 75^o = 0.258819