Evaluating the Trigonometric Expression

We need to find the value of the given trigonometric expression:

\(\frac{{\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ }}{{\cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ }}\)

Let's simplify the numerator and the denominator separately using trigonometric identities and angle properties.

Simplifying the Numerator

The numerator is \(\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ\).

We can express the angles in terms of acute angles:

• \(\cos 236^\circ = \cos (180^\circ + 56^\circ)\). Since cosine is negative in the third quadrant, \(\cos (180^\circ + \theta) = -\cos \theta\). So, \(\cos 236^\circ = -\cos 56^\circ\).
• \(\sin 124^\circ = \sin (180^\circ - 56^\circ)\). Since sine is positive in the second quadrant, \(\sin (180^\circ - \theta) = \sin \theta\). So, \(\sin 124^\circ = \sin 56^\circ\).
• We also know that \(\sin 56^\circ = \sin (90^\circ - 34^\circ)\). Using the complementary angle identity \(\sin (90^\circ - \theta) = \cos \theta\), we get \(\sin 56^\circ = \cos 34^\circ\).
• Similarly, \(\cos 56^\circ = \cos (90^\circ - 34^\circ) = \sin 34^\circ\).

Substitute these into the numerator:

\(\sin 34^\circ (-\cos 56^\circ) - \sin 56^\circ (\sin 56^\circ)\)

This does not look right. Let's use the identities derived first:

Numerator = \(\sin 34^\circ (\cos 236^\circ) - \sin 56^\circ (\sin 124^\circ)\)

Using \(\cos 236^\circ = -\cos 56^\circ\) and \(\sin 124^\circ = \sin 56^\circ\):

= \(\sin 34^\circ (-\cos 56^\circ) - \sin 56^\circ (\sin 56^\circ)\)

= \(-\sin 34^\circ \cos 56^\circ - \sin^2 56^\circ\)

Let's try substituting complementary angles differently:

Numerator = \(\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ\)

• \(\sin 56^\circ = \cos 34^\circ\)
• \(\cos 236^\circ = \cos(180^\circ + 56^\circ) = -\cos 56^\circ = -\sin 34^\circ\)
• \(\sin 124^\circ = \sin(180^\circ - 56^\circ) = \sin 56^\circ = \cos 34^\circ\)

Substitute these into the numerator:

= \(\sin 34^\circ (-\sin 34^\circ) - (\cos 34^\circ) (\cos 34^\circ)\)

= \(-\sin^2 34^\circ - \cos^2 34^\circ\)

= \(-(\sin^2 34^\circ + \cos^2 34^\circ)\)

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

= \(-(1) = -1\)

So, the numerator simplifies to \(-1\).

Simplifying the Denominator

The denominator is \(\cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ\).

Let's express the angles in a way that might lead to an identity:

• \(\cos 178^\circ = \cos (180^\circ - 2^\circ)\). Since cosine is negative in the second quadrant, \(\cos (180^\circ - \theta) = -\cos \theta\). So, \(\cos 178^\circ = -\cos 2^\circ\).
• \(\sin 208^\circ = \sin (180^\circ + 28^\circ)\). Since sine is negative in the third quadrant, \(\sin (180^\circ + \theta) = -\sin \theta\). So, \(\sin 208^\circ = -\sin 28^\circ\).

Substitute these into the denominator:

\(\cos 28^\circ \cos 88^\circ + (-\cos 2^\circ) (-\sin 28^\circ)\)

= \(\cos 28^\circ \cos 88^\circ + \cos 2^\circ \sin 28^\circ\)

Now, let's use complementary angles for \(\cos 88^\circ\) and \(\cos 2^\circ\):

• \(\cos 88^\circ = \cos (90^\circ - 2^\circ) = \sin 2^\circ\)
• \(\cos 2^\circ = \cos (90^\circ - 88^\circ) = \sin 88^\circ\)

Substitute these into the expression:

= \(\cos 28^\circ (\sin 2^\circ) + (\sin 88^\circ) \sin 28^\circ\)

= \(\sin 2^\circ \cos 28^\circ + \sin 88^\circ \sin 28^\circ\)

This form does not directly match a standard sum/difference identity like \(\sin(A+B)\) or \(\cos(A-B)\). Let's revisit the substituted expression \(\cos 28^\circ \cos 88^\circ + \cos 2^\circ \sin 28^\circ\).

Alternative approach using different angle relations:

• \(\cos 88^\circ = \cos (90^\circ - 2^\circ) = \sin 2^\circ\)
• \(\cos 178^\circ = \cos (90^\circ + 88^\circ) = -\sin 88^\circ\)
• \(\sin 208^\circ = \sin (270^\circ - 62^\circ) = -\cos 62^\circ = -\cos (90^\circ - 28^\circ) = -\sin 28^\circ\)

Substitute into the denominator:

\(\cos 28^\circ \cos 88^\circ + (-\sin 88^\circ) (-\sin 28^\circ)\)

= \(\cos 28^\circ \cos 88^\circ + \sin 88^\circ \sin 28^\circ\)

This is in the form \(\cos A \cos B + \sin A \sin B\), which is the identity for \(\cos(A-B)\). Let \(A = 88^\circ\) and \(B = 28^\circ\).

= \(\cos (88^\circ - 28^\circ)\)

= \(\cos 60^\circ\)

The value of \(\cos 60^\circ\) is \(\frac{1}{2}\).

So, the denominator simplifies to \(\frac{1}{2}\).

Final Calculation

Now we have the simplified numerator and denominator:

Numerator = \(-1\)

Denominator = \(\frac{1}{2}\)

The value of the expression is:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{-1}{1/2} = -1 \times 2 = -2\)

The value of the expression is \(-2\).

Additional Information: Angle Transformations

When simplifying trigonometric expressions involving angles outside the \(0^\circ\) to \(90^\circ\) range, it's helpful to express them in terms of acute angles using the following rules based on quadrants:

• Quadrant I (\(0^\circ\) to \(90^\circ\)): All functions are positive. Angles can be written as \(\theta\).
• Quadrant II (\(90^\circ\) to \(180^\circ\)): Sine is positive. Angles can be written as \(180^\circ - \theta\) or \(90^\circ + \theta\).
  • \(\sin(180^\circ - \theta) = \sin \theta\); \(\cos(180^\circ - \theta) = -\cos \theta\); \(\tan(180^\circ - \theta) = -\tan \theta\)
  • \(\sin(90^\circ + \theta) = \cos \theta\); \(\cos(90^\circ + \theta) = -\sin \theta\); \(\tan(90^\circ + \theta) = -\cot \theta\)
• Quadrant III (\(180^\circ\) to \(270^\circ\)): Tangent is positive. Angles can be written as \(180^\circ + \theta\) or \(270^\circ - \theta\).
  • \(\sin(180^\circ + \theta) = -\sin \theta\); \(\cos(180^\circ + \theta) = -\cos \theta\); \(\tan(180^\circ + \theta) = \tan \theta\)
  • \(\sin(270^\circ - \theta) = -\cos \theta\); \(\cos(270^\circ - \theta) = -\sin \theta\); \(\tan(270^\circ - \theta) = \cot \theta\)
• Quadrant IV (\(270^\circ\) to \(360^\circ\)): Cosine is positive. Angles can be written as \(360^\circ - \theta\) or \(270^\circ + \theta\).
  • \(\sin(360^\circ - \theta) = -\sin \theta\); \(\cos(360^\circ - \theta) = \cos \theta\); \(\tan(360^\circ - \theta) = -\tan \theta\)
  • \(\sin(270^\circ + \theta) = -\cos \theta\); \(\cos(270^\circ + \theta) = \sin \theta\); \(\tan(270^\circ + \theta) = -\cot \theta\)

Using these rules along with basic identities like \(\sin^2 \theta + \cos^2 \theta = 1\) and sum/difference formulas helps simplify complex trigonometric expressions.