How do you find the value?

Question

#cos# #alpha/2# if #tan alpha= 40/9, (180 < alpha<270)#

Elijah Alexander · Accepted Answer

#cos(alpha/2)=-4/sqrt41# Given: #(180ltalpha<270)# indicating that #cosalpha # is negative #tanalpha=40/9# opposite side is 40 adjacent side is 9 hypotenuse will be 41, because #sqrt(9^2+40^2)=sqrt(81+1600)=sqrt1681=41# #cosalpha="(adjacent side)/(hypotenuse)"# #cosalpha=-9/41# #cos(alpha/2)=+-sqrt((1+cosalpha)/2)=+-sqrt((1-9/41)/2# #=+-sqrt((41-9)/(41xx2))=+-sqrt(32/2xx1/41)=+-sqrt(16)/sqrt(41)# #cos(alpha/2)=+-4/sqrt41# As mentioned, #180^@ltalphalt270^@, # it follows that #180/2ltalpha/2<270/2# #90ltalpha/2<135#, where #cos(alpha/2)# is negative Hence, #cos(alpha/2)=-4/sqrt41#

Charles Autterson · Answer

Start with the identity:

#tan^2(alpha) +1 = sec^2(alpha)#

Substitute #sec^2(alpha)= 1/cos^2(alpha)#

#tan^2(alpha)+ 1= 1/cos^2(alpha)#

Multiply both sides by #cos^2(alpha)/(tan^2(alpha) + 1)#:

#cos^2(alpha) = 1/(tan^2(alpha) + 1)#

Use the square root operation on both sides:

#cos(alpha) = +-sqrt(1/(tan^2(alpha) + 1))#

We are told that #180^@ < alpha < 270^@#, therefore we choose the negative value:

#cos(alpha) = -sqrt(1/(tan^2(alpha) + 1))#

Add 1 to both sides:

#1+ cos(alpha) = 1-sqrt(1/(tan^2(alpha) + 1))#

Multiply both sides by #1/2#:

#(1+ cos(alpha))/2 = (1-sqrt(1/(tan^2(alpha) + 1)))/2#

Use the square root operation on both sides:

#+-sqrt((1+ cos(alpha))/2) = +-sqrt((1-sqrt(1/(tan^2(alpha) + 1)))/2)#

Substitute #+-sqrt((1+ cos(alpha))/2)= cos(alpha/2)#

#cos(alpha/2) = +-sqrt((1-sqrt(1/(tan^2(alpha) + 1)))/2)#

From #180^@ < alpha < 270^@# we derive #90^@ < alpha/2 < 135^@# and conclude that the cosine function is negative within the specified domain:

#cos(alpha/2) = -sqrt((1-sqrt(1/(tan^2(alpha) + 1)))/2)#

Substitute #tan^2(alpha) = (40/9)^2#:

#cos(alpha/2) = -sqrt((1-sqrt(1/((40/9)^2 + 1)))/2)#

I used WolframAlpha to simplify the above into an exact form:

#cos(alpha/2) = -(4sqrt41)/41#

Abigail Armstrong · Answer

undefined To find the value of something, you typically need to apply relevant mathematical operations or procedures based on the given context of the problem or question. This may involve calculations, manipulations of equations, or using known formulas or principles depending on the specific problem you're trying to solve. It's essential to carefully analyze the information provided and determine the appropriate method or approach to find the desired value accurately.