Hi Tutors,
I've tried solving the problem below, but I keep getting it wrong. I've solved 3 similar questions thanks to Reiny and Steve, but this one, in particular, is giving me problems.
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Use the given information to determine the values of sine α/2, cos α/2, and tan α/2.
tanα=15/8;π<α<3π/2
Please show how you got the answers so I can study it step by step.
As usual, make a sketch of a right-angled triangle and using Pythagoras,
r^2 = 8^2 + 15^2 = 289
r = 17
π<α<3π/2 ----> α is in quadrant III
sin α = -15/17 and cos α = -8/17
recall cos 2A = 1 - 2sin^2 A
thus: cos α = 1 - 2sin^2 (α/2)
-8/17 = 1 - 2sin^2 (α/2)
2sin^2 (α/2) = 1 + 8/17 = 25/17
sin^2 (α/2) = 25/34
sin (α/2) = 5/√34 , because (α/2) is in quadrant II
recall that cos 2A = 2cos^2 A - 1
cos α = 2cos^2 (α/2) - 1
-8/17 = 2cos^2 (α/2) - 1
1 - 8/17 = 2cos^2 (α/2)
9/17 = 2cos^2 (α/2)
9/34 = cos^2 (α/2)
cos (α/2) = -3/√34 , since the cosine is negative in II
since tan (α/2) = sin (α/2) / cos (α/2)
= (5/√34) / (-3/√34)
= - 5/3
Thank you Reiny
To find the values of sine α/2, cos α/2, and tan α/2, we need to use the given information that tan α = 15/8 and π < α < 3π/2.
First, let's find the value of tan α/2 using the half-angle formula for the tangent:
tan(α/2) = ±√((1 - cos α) / (1 + cos α))
Since we know that π < α < 3π/2, we can conclude that α is in the third quadrant, where cos α is negative. Therefore, we can select the negative square root:
tan(α/2) = -√((1 - cos α) / (1 + cos α))
Now, let's solve for cos α/2 using the half-angle formula for cosine:
cos(α/2) = ±√((1 + cos α) / 2)
Similar to before, considering that α is in the third quadrant, we choose the negative square root:
cos(α/2) = -√((1 + cos α) / 2)
Finally, we can use the Pythagorean identity to find sin(α/2):
sin(α/2) = ±√((1 - cos^2(α/2)))
Since we know cos(α/2) from the previous step, we can substitute it in:
sin(α/2) = ±√(1 - (cos(α/2))^2)
Now, let's substitute the given value of tan α = 15/8 into our calculations.
Using the given value tan α = 15/8, we can solve for α first:
tan α = 15/8
α = arctan(15/8)
Keep in mind that we need to find the values of sin α/2, cos α/2, and tan α/2, not α itself. So, let's now find α/2 using the calculated value of α.
α/2 = α / 2
Now, let's plug in the value of α we found:
α/2 = arctan(15/8) / 2
Note that the exact value of α/2 cannot be represented using simple radicals or decimal notation. It will be in terms of arctangents. However, we can use an approximation to proceed with calculating sin(α/2), cos(α/2), and tan(α/2) using a calculator.
Once you have the decimal approximation for α/2, substitute it into the formulas for sin(α/2), cos(α/2), and tan(α/2) we derived earlier to get the answers.