Question

Use the formula for the cosine of the difference of two angles to find the exact value of the following expression.

Cos(45° - 60°)

Answer 1

Exact value of Cos(45° - 60°) is 0.96 using difference of two angles.

Explanation:

Given:

Cos(45° - 60°)

We have to apply the formula of cosine for difference of the two angles.

Formula:

`cos(a-b)=cos(a)\ cos(b)+sin(a)\ sin(b)`

Plugging the values.

⇒
`cos(45-60)=cos(45)\ cos(60) + sin(45)\ sin(60)`

We know that the values :

`sin(45) =cos(45) = (1)/(√(2) )`

`sin(60)=(√(3) )/(2)` and
`cos(60)=(1)/(2)`

So,

⇒
`cos(45-60)=((1)/(√(2) ) * (1)/(2) ) + ((1)/(√(2) ) * (√(3) )/(2))`

⇒
`cos(45-60)=((1)/(2√(2) ) + (√(3) )/(2√(2) ))`

⇒
`cos(45-60)=((1+√(3) )/(2√(2) ) )`

⇒
`cos(45-60)=((1+√(3) )/(2√(2) ) )* (2√(2) )/(2√(2) )` ...rationalizing

⇒
`cos(45-60)=(2√(2) +2√(6) )/( 8)`

⇒
`cos(45-60)=(2(√(2)+√(6)))/(8)` ...taking 2 as a common factor

⇒
`cos(45-60)=((√(2)+√(6)))/(4)`

To find the exact values we have to put the values of sq-rt .

As,
`√(2)=1.41` and
`√(6) =2.44`

Then

⇒
`cos(45-60)=(( 1.41+2.44))/(4)`

⇒
`cos(45-60)=(( 3.85))/(4)`

⇒
`cos(45-60)=0.96`

So the exact value of Cos(45° - 60°) is 0.96 using difference of two angles.