1) Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.

67 degrees 30'

To use the half-angle formulas, we first need to convert the given angle of 67 degrees 30' into radians.

To convert degrees to radians, we use the formula:

Radian = (pi/180) * Degree

Let's calculate the radian measure of the given angle:

Radian = (pi/180) * 67.5

Radian ≈ 1.1781

Now, we can use the half-angle formulas:

1) Sine (θ/2) = ± √((1 - cosθ) / 2)

2) Cosine (θ/2) = ± √((1 + cosθ) / 2)

3) Tangent (θ/2) = sin(θ) / (1 + cos(θ))

Let's substitute θ = 67.5 degrees (or 1.1781 radians) into these formulas:

1) Sine (67.5/2):
Sine (67.5/2) = ± √((1 - cos(67.5)) / 2)

2) Cosine (67.5/2):
Cosine (67.5/2) = ± √((1 + cos(67.5)) / 2)

3) Tangent (67.5/2):
Tangent (67.5/2) = sin(67.5) / (1 + cos(67.5))

To find the exact values, we need to evaluate the cosine and sine of 67.5 degrees. We can use either a calculator or mathematical tables to find these values.

Once we have the values of cos(67.5) and sin(67.5), we can substitute them into the formulas above to calculate the exact values of the sine, cosine, and tangent of the angle 67 degrees 30'.