Question

2cos 10+ sin100 +sin1000 + sin 10000 simplifies to ???

A) cos10
B) 3cos10
C) 4cos10
D) 5cos10​

Answer 1

$\huge\text{\underline{\underline{Question}}}$

What value does $2\cos10^{\circ}+\sin100^{\circ}+\sin1000^{\circ}+\sin10000^{\circ}$ simplify to?

$\huge\text{\underline{\underline{Explanation}}}$

$\sin100^{\circ}$

$=\sin(180^{\circ}-80^{\circ})$

$=\sin80^{\circ}$

$=\sin(90^{\circ}-10^{\circ})$

$\text{ =\boxed{\cos10^{\circ}} }$

$\sin1000^{\circ}$

$=\sin(360^{\circ}\times3-80)^{\circ}$

$=\sin(360^{\circ}-80)^{\circ}$

$=-\sin80^{\circ}$

$=-\sin(90^{\circ}-10^{\circ})$

$\text{ =\boxed{-\cos10^{\circ}} }$

$\sin10000^{\circ}$

$=\sin(360^{\circ}\times27-80)^{\circ}$

$=\sin(360^{\circ}-80)^{\circ}$

$=-\sin80^{\circ}$

$=-\sin(90^{\circ}-10^{\circ})$

$\text{ =\boxed{-\cos10^{\circ}} }$

$\bold{Now,\ we\ know\ that\ -}$

$\bullet\ \sin100^{\circ}=\cos10^{\circ}$

$\bullet\ \sin1000^{\circ}=-\cos10^{\circ}$

$\bullet\ \sin10000^{\circ}=-\cos10^{\circ}$

$\huge\text{\underline{\underline{Final answer}}}$

$\bold{Finally,\ -}$

$\text{ \cdots\longrightarrow\boxed{2\cos10^{\circ}+\sin100^{\circ}+\sin1000^{\circ}+\sin10000^{\circ}=\cos10^{\circ}} }$

$\bold{Option\ A\ is\ correct.}$

$\huge\text{\underline{\underline{Helpful guide}}}$

$\Large\text{\{Radian\}}$

$\large\text{ \longrightarrow Converting degree to radian}$

$\text{ \bullet\ \pi rad =180^{\circ} }$

$\Large\text{\{Trigonometric functions\}}$

$\large\text{ \longrightarrow Periods of functions}$

$\bullet\ \sin(\theta+2\pi)=\sin\theta$

$\bullet\ \cos(\theta+2\pi)=\cos\theta$

$\bullet\ \tan(\theta+\pi)=\tan\theta$

$\Large\text{\{Functions on the unit circle\}}$

$\large\text{ \longrightarrow x^{2}+y^{2}=1 (Unit circle)}$

$\bullet\ \sin\theta=y$

$\bullet\ \cos\theta=x$

$\bullet\ \tan\theta=\dfrac{y}{x}$

$\Large\text{\{Sign changes\}}$

$\large\text{ \longrightarrow Positive signs in each quadrant}$

$\text{ \bullet 1st quadrant: All}$

$\text{ \bullet 2nd quadrant: \sin\theta>0 }$

$\text{ \bullet 3rd quadrant: \tan\theta>0 }$

$\text{ \bullet 4th quadrant: \cos\theta>0 }$

$\Large\text{\{Reciprocal\}}$

$\large\text{ \longrightarrow Reciprocal functions}$

$\bullet\ \sin\theta\csc\theta=1$

$\bullet\ \tan\theta\cot\theta=1$

$\bullet\ \cos\theta\sec\theta=1$

$\Large\text{\{Identity\}}$

$\large\text{ \longrightarrow Trigonometric identities}$

$\bullet\ \sin^{2}\theta+\cos^{2}\theta=1$

$\bullet\ \tan^{2}\theta+1=\sec^{2}\theta$

Answer 2

• please check the attached file