sec( pi/7)-sec(2pi/7)+sec(3pi/7)
To find the value of sec(pi/7) - sec(2pi/7) + sec(3pi/7), we need to recall the following relationship:
sec(x) = 1/cos(x)
Using this relationship, we can rewrite the expression as:
1/cos(pi/7) - 1/cos(2pi/7) + 1/cos(3pi/7)
Now, let's simplify each term individually.
1/cos(pi/7):
To evaluate this term, we need to find the value of cos(pi/7). We can use the unit circle or a calculator to determine that cos(pi/7) ≈ 0.6235. Thus,
1/cos(pi/7) = 1/0.6235 ≈ 1.6040
1/cos(2pi/7):
Similarly, we need to determine the value of cos(2pi/7). Again, we can use the unit circle or a calculator to find that cos(2pi/7) ≈ 0.2225. Therefore,
1/cos(2pi/7) = 1/0.2225 ≈ 4.4956
1/cos(3pi/7):
Finally, we find cos(3pi/7) ≈ -0.2225. Therefore,
1/cos(3pi/7) = 1/(-0.2225) ≈ -4.4956
Now, we substitute these values back into the original expression:
1.6040 - 4.4956 + (-4.4956)
Which simplifies to:
-7.3872
So, sec(pi/7) - sec(2pi/7) + sec(3pi/7) is approximately equal to -7.3872.