Geometry

Pythagorean Identities

Pythagorean Identities: Level 2 Challenges

         

Simplify: 1sin2θ. \sqrt{1 - \sin^2 \theta}.

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.

Letting P=sinAsinBQ=sinCcosAR=sinAcosBS=cosAcosCP=\sin A \sin B \\ Q= \sin C \cos A \\ R = \sin A \cos B \\ S=\cos A \cos C

Find the value of 5(P2+Q2+R2+S2)5(P^2+Q^2+R^2+S^2) .

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?

A triangle has sides of magnitude 11, sinx \sin x, and cosx\cos x.

where 0<x<π2.0 < x < \frac { \pi }{ 2 } .

Find the largest angle of the triangle in degrees.

Assume that the triangle is non-degenerate.

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