Geometry

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities: Level 3 Challenges

         

sin21+sin22+sin23++sin288+sin289+sin290= ? \sin ^{ 2 }{ 1^{\circ} } +\sin ^{ 2 }{ 2^{\circ} } +\sin ^{ 2 }{ 3^{\circ} } + \ldots \\ +\sin ^{ 2 }{ 88^{\circ} } +\sin ^{ 2 }{ 89^{\circ} } +\sin ^{ 2 }{ 90^{\circ} } = \ ?

sin2θ5=cos2θ6 \large \frac { \sin ^{ 2 }{ \theta } }{ 5 } =\frac { \cos ^{ 2 }{ \theta } }{ 6 }

If θ\theta is a positive acute angle that satisfies the equation above, find sinθ\sin { \theta }.

Note: Give your answer to 3 decimal places.

k=150[(1+tan(k)+sec(k)) (1+cot(k)csc(k))]= ?  \displaystyle \sum_{k=1}^{50} \Bigg [ \bigg(1 + \tan(k^\circ)+\sec(k^\circ)\bigg)\ \bigg(1+\cot(k^\circ)-\csc(k^\circ) \bigg)\Bigg]=\ ? \

(2+2)x+(22)x=2x \Large \left(\sqrt{2+\sqrt{2}}\right)^{x} + \left(\sqrt{2-\sqrt{2}}\right)^{x} = 2^{x}

Find the sum of all real xx that satisfy the equation above.

1cos2θ+11+sin2θ+21+sin4θ+41+sin8θ \large\frac { 1 }{ \cos ^{ 2 }{ \theta } } +\frac { 1 }{ 1+\sin ^{ 2 }{ \theta } } +\frac { 2 }{ 1+\sin ^{ 4 }{ \theta } } +\frac { 4 }{ 1+\sin ^{ 8 }{ \theta } }

If sin16θ=15 \large \sin ^{ 16 }{ \theta } = \frac { 1 }{ 5 } , what is the value of the expression above?

×

Problem Loading...

Note Loading...

Set Loading...