1) Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.
67 degrees 30'
Use the half angle formulas at
http://www.sosmath.com/trig/douangl/douangl.html
The sine of the full angle (135 deg) is 1/sqrt2 and the cosing is -1/sqrt2
To use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle, follow these steps:
Step 1: Convert the angle to radians
To use the half-angle formulas, the angle needs to be in radians. Convert 67 degrees 30' to radians using the following conversion:
67 degrees 30' = (67 + 30/60) * π/180 ≈ 1.1781 radians
Step 2: Apply the half-angle formulas
The half-angle formulas are as follows:
sin(x/2) = ±√[(1 - cos(x))/2]
cos(x/2) = ±√[(1 + cos(x))/2]
tan(x/2) = sin(x)/(1 + cos(x))
Using these formulas, we can find the values of sine, cosine, and tangent for the angle.
Step 3: Calculate sine (sin(x/2))
sin(x/2) = ±√[(1 - cos(x))/2]
= ±√[(1 - cos(1.1781))/2]
To determine the exact value, you can use a calculator or lookup table to find the value of cos(1.1781) and substitute it into the formula.
Step 4: Calculate cosine (cos(x/2))
cos(x/2) = ±√[(1 + cos(x))/2]
= ±√[(1 + cos(1.1781))/2]
Again, use a calculator or lookup table to find the value of cos(1.1781) and substitute it into the formula to obtain the exact value.
Step 5: Calculate tangent (tan(x/2))
tan(x/2) = sin(x)/(1 + cos(x))
= sin(1.1781)/(1 + cos(1.1781))
Use a calculator or lookup table to find the value of sin(1.1781) and cos(1.1781), substitute them into the formula, and calculate the exact value of tangent.
By following these steps and substituting the appropriate values, you can determine the exact values of the sine, cosine, and tangent of the given angle.