Trigonometry Formulae - Part 7

Algebra Level pending

In $\displaystyle △ABC$ , $\displaystyle C>90°$ , $\displaystyle E=sinC$ , $\displaystyle F=sinA+sinB$ , $\displaystyle G=cosA+cosB$ .

What is the relationship between $\displaystyle E$ , $\displaystyle F$ , $\displaystyle G$ ?

\displaystyle F>G>E

\displaystyle E>F>G

\displaystyle G>F>E

\displaystyle F>E>G

1 solution

A Former Brilliant Member
May 19, 2019

Since C> $\dfrac{π}{2}$ , therefore A+B< $\dfrac{π}{2}$ . Hence $\dfrac{A+B}{2}$ < $\dfrac{π}{4}$ and tan( $\dfrac{A+B}{2}$ )<1. Hence sin( $\dfrac{A+B}{2}$ )<cos( $\dfrac{A+B}{2}$ ), or sin( $\dfrac{C}{2}$ )<cos( $\dfrac{C}{2}$ ). Also cos( $\dfrac{A+B}{2}$ )<cos( $\dfrac{A-B}{2}$ ). This implies G>F>E

Trigonometry Formulae - Part 7

1 solution

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