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Find the exact value of cos 105⁰.

a. √√√2-√6
4
b.
√2+√6
4
C.
4
d. √2+√6
4

1 Answer

5 votes

Answer:


(√(2)-√(6) )/(4) }

Explanation:

Find the exact value of cos(105°).

The method I am about to show you will allow you to complete this problem without a calculator. Although, memorizing the trigonometric identities and the unit circle is required.

We have,


\cos(105\°)

Using the angle sum identity for cosine.


\boxed{\left\begin{array}{ccc}\text{\underline{Angle Sum Identity for Cosine}}\\\\\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\end{array}\right}

Split the given angle, in degrees, into two angles. Preferably two angles we can recognize on the unit circle.


105\textdegree=45\textdegree+60\textdegree\\\\\\\therefore \cos(105\textdegree)=\cos(45\textdegree+60\textdegree)

Now applying the identity.


\cos(45\textdegree+60\textdegree)\\\\\\\Longrightarrow \cos(45\textdegree+60\textdegree)=\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)

Now utilizing the unit circle.


\boxed{\left\begin{array}{ccc}\text{\underline{From the Unit Circle:}}\\\\\cos(45\textdegree)=(√(2) )/(2)\\\\\cos(60\textdegree)=(1)/(2)\\\\\sin(45\textdegree)=(√(2) )/(2)\\\\\sin(60\textdegree)=(√(3) )/(2) \end{array}\right}


\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)\\\\\\\Longrightarrow \Big((√(2) )/(2)\Big)\Big((1 )/(2)\Big)-\Big((√(2) )/(2)\Big)((√(3) )/(2)\Big)

Now simplifying...


\Big((√(2) )/(2)\Big)\Big((1 )/(2)\Big)-\Big((√(2) )/(2)\Big)((√(3) )/(2)\Big)\\\\\\\Longrightarrow \Big((√(2) )/(4) \Big)-\Big((√(6) )/(4) \Big)\\\\\\\therefore \cos(105\textdegree)= \boxed{\boxed{(√(2)-√(6) )/(4) }}

Find the exact value of cos 105⁰. a. √√√2-√6 4 b. √2+√6 4 C. 4 d. √2+√6 4-example-1
User Mike Housky
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