Question

Evaluate :cos 45 by sin 30 + cosec 30​

Answer 1

see explanation

Explanation:

Using the identity

cosecx =
`(1)/(sinx)` and the exact values

cos45° =
`(√(2) )/(2)`, sin30° =
`(1)/(2)`

Given

cos45° × sin30° + cosec30°

=
`(√(2) )/(2)` ×
`(1)/(2)` +
`(1)/((1)/(2) )`

=
`(√(2) )/(4)` + 2

=
`(√(2) )/(4)` +
`(8)/(4)`

=
`(1)/(4)`(
`√(2)` + 8 ) ← exact value

≈2.354 ( 3 dec. places )

Answer 2

`\large\boxed{2+\sqrt2}\ or\ \boxed{(8+\sqrt2)/(4)}`

Explanation:

We know:

`\csc\theta=(1)/(\sin\theta)`

From the table (attachment):

`\cos45^o=(\sqrt2)/(2)\\\\\sin30^o=(1)/(2)\\\\\csc30^o=(1)/(\sin30^o)=(1)/((1)/(2))=1\cdot(2)/(1)=2`

Substitute:

If is:

`(\cos45^o)/(\sin30^o)+\csc30^o=((\sqrt2)/(2))/((1)/(2))+2=(\sqrt2)/(2)\cdot(2)/(1)+2=\sqrt2+2`

If is:

`\cos45^o\cdot\sin30^o+\csc30^o=(\sqrt2)/(2)\cdot(1)/(2)+2=(\sqrt2)/(4)+2=(\sqrt2)/(4)+(8)/(4)=(8+\sqrt2)/(4)`