Question

Evaluate :cos 45 by sin 30 + cosec 30​

Answer 1

Answer:

see explanation

Step-by-step explanation:

Using the identity

cosecx = $\frac{1}{sinx}$ and the exact values

cos45° = $\frac{\sqrt{2} }{2}$, sin30° = $\frac{1}{2}$

Given

cos45° × sin30° + cosec30°

= $\frac{\sqrt{2} }{2}$ × $\frac{1}{2}$ + $\frac{1}{\frac{1}{2} }$

= $\frac{\sqrt{2} }{4}$ + 2

= $\frac{\sqrt{2} }{4}$ + $\frac{8}{4}$

= $\frac{1}{4}$($\sqrt{2}$ + 8 ) ← exact value

≈2.354 ( 3 dec. places )

Answer 2

Answer:

$\large\boxed{2+\sqrt2}\ or\ \boxed{\dfrac{8+\sqrt2}{4}}$

Step-by-step explanation:

We know:

$\csc\theta=\dfrac{1}{\sin\theta}$

From the table (attachment):

$\cos45^o=\dfrac{\sqrt2}{2}\\\\\sin30^o=\dfrac{1}{2}\\\\\csc30^o=\dfrac{1}{\sin30^o}=\dfrac{1}{\frac{1}{2}}=1\cdot\dfrac{2}{1}=2$

Substitute:

If is:

$\dfrac{\cos45^o}{\sin30^o}+\csc30^o=\dfrac{\frac{\sqrt2}{2}}{\frac{1}{2}}+2=\dfrac{\sqrt2}{2}\cdot\dfrac{2}{1}+2=\sqrt2+2$

If is:

$\cos45^o\cdot\sin30^o+\csc30^o=\dfrac{\sqrt2}{2}\cdot\dfrac{1}{2}+2=\dfrac{\sqrt2}{4}+2=\dfrac{\sqrt2}{4}+\dfrac{8}{4}=\dfrac{8+\sqrt2}{4}$