If \( \cos ^{2} 23^{\circ}=t \), determine the following in terms of \( t \) : (1) \( \sin ^{2} 337^{\circ} \) (2) \( \sin 23^{\circ} \) (3) \( \cos 46^{\circ} \) (4) \( \sin 46^{\circ} \) (5) \( \tan \left(-226^{\circ}\right) \) (6) \( \sin 37^{\circ} \)

To solve the problem, we will use the known relationships between trigonometric functions and the given value \( t = \cos^2 23^\circ \). ### Step 1: Determine \( \sin^2 337^\circ \) Using the identity \( \sin(360^\circ - x) = -\sin(x) \): \[ \sin 337^\circ = \sin(360^\circ - 23^\circ) = -\sin 23^\circ \] Thus, \[ \sin^2 337^\circ = (-\sin 23^\circ)^2 = \sin^2 23^\circ \] Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \): \[ \sin^2 23^\circ = 1 - \cos^2 23^\circ = 1 - t \] ### Step 2: Determine \( \sin 23^\circ \) Using the identity \( \sin^2 x + \cos^2 x = 1 \): \[ \sin 23^\circ = \sqrt{1 - \cos^2 23^\circ} = \sqrt{1 - t} \] ### Step 3: Determine \( \cos 46^\circ \) Using the double angle formula \( \cos(2x) = 2\cos^2(x) - 1 \): \[ \cos 46^\circ = \cos(2 \times 23^\circ) = 2\cos^2 23^\circ - 1 = 2t - 1 \] ### Step 4: Determine \( \sin 46^\circ \) Using the identity \( \sin(90^\circ - x) = \cos x \): \[ \sin 46^\circ = \cos(90^\circ - 46^\circ) = \cos 44^\circ \] Using the identity \( \cos(90^\circ - x) = \sin x \): \[ \sin 46^\circ = \sin(90^\circ - 46^\circ) = \sin 44^\circ \] Using the sine addition formula: \[ \sin 44^\circ = \sin(23^\circ + 23^\circ) = 2\sin 23^\circ \cos 23^\circ = 2\sqrt{1 - t} \sqrt{t} = 2\sqrt{t(1 - t)} \] ### Step 5: Determine \( \tan(-226^\circ) \) Using the periodicity of the tangent function: \[ \tan(-226^\circ) = \tan(360^\circ - 226^\circ) = \tan(134^\circ) \] Using the identity \( \tan(180^\circ - x) = -\tan x \): \[ \tan(134^\circ) = -\tan(46^\circ) \] Using the identity \( \tan x = \frac{\sin x}{\cos x} \): \[ \tan(46^\circ) = \frac{\sin 46^\circ}{\cos 46^\circ} = \frac{2\sqrt{t(1 - t)}}{2t - 1} \] Thus, \[ \tan(-226^\circ) = -\frac{2\sqrt{t(1 - t)}}{2t - 1} \] ### Step 6: Determine \( \sin 37^\circ \) Using the identity \( \sin(90^\circ - x) = \cos x \): \[ \sin 37^\circ = \cos(90^\circ - 37^\circ) = \cos 53^\circ \] Using the identity \( \cos(90^\circ - x) = \sin x \): \[ \sin 37^\circ = \sin(90^\circ - 53^\circ) = \sin 53^\circ \] Using the sine addition formula: \[ \sin 53^\circ = \sin(23^\circ + 30^\circ) = \sin 23^\circ \cos 30^\circ + \cos 23^\circ \sin 30^\circ \] Substituting known values: \[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \] Thus, \[ \sin 53^\circ = \sqrt{1 - t} \cdot \frac{\sqrt{3}}{2} + \sqrt{t} \cdot \frac{1}{2} \] ### Summary of Results 1. \( \sin^2 337^\circ = 1 - t \) 2. \( \sin 23^\circ = \sqrt{1 - t} \) 3. \( \cos 46^\circ = 2t - 1 \) 4. \( \sin 46^\circ = 2\sqrt{t(1 - t)} \) 5. \( \tan(-226^\circ) = -\frac{2\sqrt{t(1 - t)}}{2t - 1} \) 6. \( \sin 37^\circ = \sqrt{1 - t} \cdot \frac{\sqrt{3}}{2} + \sqrt{t} \cdot \frac{1}{2} \)