SOLUTION: Use a half - angle identity to find the exact value of each expression. 5) tan 45° 6) sin 165° 7) sin 5π —— 6 8.) cos 30°

Algebra ->  Finance -> SOLUTION: Use a half - angle identity to find the exact value of each expression. 5) tan 45° 6) sin 165° 7) sin 5π —— 6 8.) cos 30°       Log On


   

Question 1194933: Use a half - angle identity to find the exact value of each expression.
5) tan 45°
6) sin 165°
7) sin 5π
——
6
8.) cos 30°
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

(5)  tan(45°) = 1.   It is a table value,

                     which means that any/every/each student must know the answer from 
                     the table, without applying half-angle identity or any other identity.



(7)  sin(5π) = 0.   Every student must know it without applying half-angle identity or any other identity.




(8)  cos(30°) = sqrt%283%29%2F2.   It is a table value,

                     which means that any/every/each student must know the answer from 
                     the table, without applying half-angle identity or any other identity.


May I complete this post by joking ?

        To deduce / (to derive) the values of given angles using half-angle formulas
        is the same as to deduce multiplication table from the driving rules.



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Use a half - angle identity to find the exact value of each expression.
so, identities you need are:
sin%28theta%2F2%29=sqrt%28%281-cos%28theta%29%29%2F2%29
cos%28theta%2F2%29=sqrt%28%281%2Bcos%28theta%29%29%2F2%29
tan%28theta%2F2%29=sin%28theta%29%2F%28cos%28theta%29+%2B+1%29


5) tan+45%C2%B0
theta%2F2=45° =>theta=90°
tan%2845%29=sin%2890%29%2F%28cos%2890%29+%2B+1%29........sin%2890%29=1, cos%2890%29+=0
tan%2845%29=1%2F%280+%2B+1%29
tan+%2845%29=+1


6) sin+165°
%28theta%2F2%29=165° =>theta=330

sin%28165%29=sqrt%28%281-cos%28330%29%29%2F2%29

sin%28165%29=sqrt%28%281-sqrt%283%29%2F2%29%2F2%29

sin%28165%29=sqrt%28%28%281%2F2%29+%282+-+sqrt%283%29%29%29%2F2%29
sin%28165%29=%28sqrt%283%29+-+1%29%2F%282+sqrt%282%29%29


7) sin+%285pi%2F6%29

theta%2F2=%285pi%2F6%29 =>theta=2%285pi%2F6%29+=%285pi%2F3%29+

sin%285pi%2F6%29=sqrt%28%281-cos%285pi%2F3%29%29%2F2%29

sin%285pi%2F6%29=sqrt%28%281%2F2%29%2F2%29

sin%285pi%2F6%29=sqrt%281%2F4%29

sin%285pi%2F6%29=1%2F2


8. cos%2830%29°
theta%2F2=30 =>theta=60+

cos%2830%29=sqrt%28%281%2Bcos%2860%29%29%2F2%29

cos%2830%29=sqrt%28%281%2B1%2F2%29%2F2%29

cos%2830%29=sqrt%283%2F4%29

cos%2830%29=sqrt%283%29%2F2