Final answer:
To find cos(α - β), we use the cosine sum formula along with the Pythagorean identity to find the missing sine and cosine values for α and β, then plug them into the formula to get cos(α - β) = (4√17 - 56)/81.
Step-by-step explanation:
To find cos(α - β), we will use the cosine sum and difference formula which is cos(α ± β) = cos α cos β ± sin α sin β. Given that cos(α) = -√17/9 and α is in the second quadrant and sin(β) = -7/9 with β in the third quadrant, we know that:
- The cosine of any angle in the second quadrant is negative, matching our given cos α.
- The sine of any angle in the third quadrant is also negative, which is consistent with sin β.
We also need to find the corresponding sine and cosine values for the opposite angles. Since sine is the y-coordinate on the unit circle, and we are given cos α, we can use the Pythagorean identity sin2 α + cos2 α = 1 to find sin α. Similarly, find cos β from sin β using the identity.
sin α = ±√(1 - cos2 α) = ±√(1 - (-√17/9)2) = ±√(1 - 17/81) = ±√(64/81) = 8/9
Since α is in the second quadrant, sin α > 0, so sin α = 8/9.
cos β = ±√(1 - sin2 β) = ±√(1 - (-7/9)2) = ±√(1 - 49/81) = ±√(32/81) = -4/9
Because β is in the third quadrant, cos β < 0, so cos β = -4/9.
Now we can plug these values into the cosine sum formula:
cos(α - β) = cos α cos β + sin α sin β = (-√17/9)(-4/9) + (8/9)(-7/9) = (4√17/81) - (56/81).
After simplifying the expression, we get:
cos(α - β) = (4√17 - 56)/81.