Answer:
see explanation
Step-by-step explanation:
Using the identity
cosecx = [tex]\frac{1}{sinx}[/tex] and the exact values
cos45° = [tex]\frac{\sqrt{2} }{2}[/tex], sin30° = [tex]\frac{1}{2}[/tex]
Given
cos45° × sin30° + cosec30°
= [tex]\frac{\sqrt{2} }{2}[/tex] × [tex]\frac{1}{2}[/tex] + [tex]\frac{1}{\frac{1}{2} }[/tex]
= [tex]\frac{\sqrt{2} }{4}[/tex] + 2
= [tex]\frac{\sqrt{2} }{4}[/tex] + [tex]\frac{8}{4}[/tex]
= [tex]\frac{1}{4}[/tex]([tex]\sqrt{2}[/tex] + 8 ) ← exact value
≈2.354 ( 3 dec. places )
Answer:
[tex]\large\boxed{2+\sqrt2}\ or\ \boxed{\dfrac{8+\sqrt2}{4}}[/tex]
Step-by-step explanation:
We know:
[tex]\csc\theta=\dfrac{1}{\sin\theta}[/tex]
From the table (attachment):
[tex]\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[/tex]
Substitute:
If is:
[tex]\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[/tex]
If is:
[tex]\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}[/tex]
