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# Pythagorean Identities

Trigonometric identities bring new life to the Pythagorean theorem by re-envisioning the legs of a right triangle as sine and cosine.

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

\[ \large \frac { \sin ^{ 2 }{ \theta } }{ 5 } =\frac { \cos ^{ 2 }{ \theta } }{ 6 }\]

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

**Note:** Give your answer to 3 decimal places.

Letting \[P=\sin A \sin B \\ Q= \sin C \cos A \\ R = \sin A \cos B \\ S=\cos A \cos C\]

Find the value of \(5(P^2+Q^2+R^2+S^2)\) .

\[ \large\frac { 1 }{ \cos ^{ 2 }{ \theta } } +\frac { 1 }{ 1+\sin ^{ 2 }{ \theta } } +\frac { 2 }{ 1+\sin ^{ 4 }{ \theta } } +\frac { 4 }{ 1+\sin ^{ 8 }{ \theta } } \]

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

A triangle has sides of magnitude \(1\), \( \sin x\), and \(\cos x\).

where \(0 < x < \frac { \pi }{ 2 } .\)

Find the largest angle of the triangle in degrees.

Assume that the triangle is non-degenerate.

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