Find the exact value of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formula, given that cscu = -5/3 and π < u < 3π/2
π < u < 3 π / 2
180° < u < 270°
u lies in Quadrant III
csc ( u ) = 1 / sin ( u ) = - 5 / 3
sin ( u ) = - 3 / 5
cos ( u ) = ±√ ( 1 - sin² u )
cos ( u ) = ±√ [ 1 - ( - 3 / 5 )² ]
cos ( u ) = ±√ ( 1 - 9 / 25 )
cos ( u ) = ±√ ( 25 / 25 - 9 / 25 )
cos ( u ) = ±√ ( 16 / 25 )
cos ( u ) = ± 4 / 5
in Quadrant III cosine is negative so:
cos ( u ) = - 4 / 5
π / 2 < u / 2 < 3 π / 4
90° < u / 2 < 135°
u / 2 lies in Quadrant II
sin ( u / 2 ) = ±√ [ ( 1 - cos u ) / 2 ]
sin ( u / 2 ) = ±√ [ ( 1 - ( - 4 / 5 ) ) / 2 ]
sin ( u / 2 ) = ±√ [ ( 1 + 4 / 5 ) / 2 ]
sin ( u / 2 ) = ±√ [ ( 5 / 5 + 4 / 5 ) / 2 ]
sin ( u / 2 ) = ±√ [ ( 9 / 5 ) / 2 ]
sin ( u / 2 ) = ±√ ( 9 / 10 )
sin ( u / 2 ) = ±√ 9 / √ 10
sin ( u / 2 ) = ± 3 / √10
in Quadrant II sine is postive so:
sin ( u / 2 ) = 3 / √10
cos ( u / 2 ) = ±√ [ ( 1 + cos u ) / 2 ]
cos ( u / 2 ) = ±√ [ ( 1 + ( - 4 / 5 ) ) / 2 ]
cos ( u / 2 ) = ±√ [ ( 1 - 4 / 5 ) / 2 ]
cos ( u / 2 ) = ±√ [ ( 5 / 5 - 4 / 5 ) / 2 ]
cos ( u / 2 ) = ±√ [ ( 1 / 5 ) / 2 ]
cos ( u / 2 ) = ±√ ( 1 / 10 )
cos ( u / 2 ) = ±√ 1 / √ 10
cos ( u / 2 ) = ± 1 / √10
in Quadrant II cosine is negative so:
cos ( u / 2 ) = - 1 / √10
tan ( u / 2 ) = ±√ [ ( 1 - cos u ) / ( 1 + cos u ) ]
tan ( u / 2 ) = ±√ [ ( 1 - ( - 4 / 5 ) ) / ( 1 + ( - 4 / 5 ) ]
tan ( u / 2 ) = ±√ [ ( 1 + 4 / 5 ) / ( 1 - 4 / 5 ) ]
tan ( u / 2 ) = ±√ [ ( 5 / 5 + 4 / 5 ) / ( 5 / 5 - 4 / 5 ) ]
tan ( u / 2 ) = ±√ [ ( 9 / 5 ) / ( 1 / 5 ) ]
tan ( u / 2 ) = ±√ 9
tan ( u / 2 ) = ± 3
in Quadrant II tangent is negative so:
tan ( u / 2 ) = - 3
Thank you very much
To find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formula, we need to first find the value of sin(u) or cos(u) by using the given information.
Given that cscu = -5/3 and π < u < 3π/2, we can start by using the reciprocal identity of the sine function.
cscu = 1/sinu
Therefore, we can rewrite the equation as:
-5/3 = 1/sinu
Now, we can solve for sinu by taking the reciprocal of both sides:
sinu = -3/5
Next, we need to determine if sinu is positive or negative in the given range. Since π < u < 3π/2, which is in the third quadrant of the unit circle, sinu will be negative. Thus, sinu = -3/5.
Now, let's apply the half-angle formula to find the exact values of sin(u/2), cos(u/2), and tan(u/2):
sin(u/2) = ±√[(1 - cosu)/2]
cos(u/2) = ±√[(1 + cosu)/2]
tan(u/2) = sin(u/2)/cos(u/2)
In our case, sinu = -3/5, and since sinu is negative, we take the negative square root.
sin(u/2) = -√[(1 - cosu)/2]
Now, let's find the value of cosu:
cos^2(u) + sin^2(u) = 1
cos^2(u) + (-3/5)^2 = 1
cos^2(u) + 9/25 = 1
cos^2(u) = 16/25
cosu = ±4/5
Since u is in the third quadrant, cosu will be negative. Therefore, cosu = -4/5.
Now, substitute the values of sinu and cosu into the half-angle formulas:
sin(u/2) = -√[(1 - cosu)/2]
= -√[(1 - (-4/5))/2]
= -√[(1 + 4/5)/2]
= -√[(9/5)/2]
= -√(9/10)
= -3/√10
cos(u/2) = ±√[(1 + cosu)/2]
= ±√[(1 + (-4/5))/2]
= ±√[(1 - 4/5)/2]
= ±√[(1/5)/2]
= ±√(1/10)
= ±1/√10
tan(u/2) = sin(u/2)/cos(u/2)
= (-3/√10)/(±1/√10)
= (-3/√10)*(√10/±1)
= -3/±1
= -3 or 3
Therefore, the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formula are:
sin(u/2) = -3/√10
cos(u/2) = ±1/√10
tan(u/2) = -3 or 3