Evaluating Trigonometric Expressions: Step-by-Step Solution

We need to find the value of the given trigonometric expression: \( \sin^4 30^\circ + \cos^4 30^\circ - \sin 25^\circ \cos 65^\circ - \sin 65^\circ \cos 25^\circ \).

Let's break down the expression into two main parts and evaluate each part separately.

Part 1: Evaluating \( \sin^4 30^\circ + \cos^4 30^\circ \)

First, recall the standard trigonometric values for \(30^\circ\):

• \( \sin 30^\circ = \frac{1}{2} \)
• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

Now, let's calculate the fourth powers:

• \( \sin^4 30^\circ = (\sin 30^\circ)^4 = \left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16} \)
• \( \cos^4 30^\circ = (\cos 30^\circ)^4 = \left(\frac{\sqrt{3}}{2}\right)^4 = \frac{(\sqrt{3})^4}{2^4} = \frac{(\sqrt{3}^2)^2}{16} = \frac{3^2}{16} = \frac{9}{16} \)

Adding these two values:

\( \sin^4 30^\circ + \cos^4 30^\circ = \frac{1}{16} + \frac{9}{16} = \frac{1+9}{16} = \frac{10}{16} \)

This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2:

\( \frac{10}{16} = \frac{10 \div 2}{16 \div 2} = \frac{5}{8} \)

So, the first part of the expression evaluates to \( \frac{5}{8} \).

Part 2: Evaluating \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ \)

The second part of the expression is \( -(\sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ) \). Let's evaluate the term inside the parenthesis.

This term looks like a familiar trigonometric identity. Recall the sine addition formula:

\( \sin(A+B) = \sin A \cos B + \cos A \sin B \)

Comparing this with our term, \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ \), we can see that it matches the right side of the identity with \( A = 25^\circ \) and \( B = 65^\circ \). Note that the order of the second term is reversed in the original expression (\( \sin 65^\circ \cos 25^\circ \) instead of \( \cos 25^\circ \sin 65^\circ \)), but multiplication is commutative, so \( \sin 65^\circ \cos 25^\circ = \cos 25^\circ \sin 65^\circ \).

Using the identity:

\( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ = \sin(25^\circ + 65^\circ) = \sin 90^\circ \)

We know that \( \sin 90^\circ = 1 \).

Alternatively, we could use complementary angle identities. Since \( 25^\circ + 65^\circ = 90^\circ \), \( 65^\circ \) is the complement of \( 25^\circ \). Thus:

• \( \cos 65^\circ = \cos(90^\circ - 25^\circ) = \sin 25^\circ \)
• \( \sin 65^\circ = \sin(90^\circ - 25^\circ) = \cos 25^\circ \)

Substituting these into the term \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ \):

\( \sin 25^\circ (\sin 25^\circ) + (\cos 25^\circ) \cos 25^\circ = \sin^2 25^\circ + \cos^2 25^\circ \)

Recall the fundamental trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \). With \( \theta = 25^\circ \), we have:

\( \sin^2 25^\circ + \cos^2 25^\circ = 1 \)

Both methods confirm that \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ = 1 \).

So, the second part of the original expression, \( -(\sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ) \), evaluates to \( -(1) = -1 \).

Combining the Parts to Find the Final Value

The original expression is the sum of the results from Part 1 and the negative of the result from Part 2:

Expression Value = (Value of \( \sin^4 30^\circ + \cos^4 30^\circ \)) - (Value of \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ \))

Expression Value = \( \frac{5}{8} - 1 \)

To subtract 1 from \( \frac{5}{8} \), we write 1 as a fraction with a denominator of 8:

\( 1 = \frac{8}{8} \)

So, Expression Value = \( \frac{5}{8} - \frac{8}{8} = \frac{5-8}{8} = \frac{-3}{8} \)

The final value of the expression is \( -\frac{3}{8} \).

Summary of Evaluation Steps

Here is a summary of the steps taken to evaluate the expression:

• Identified the expression components.
• Evaluated \( \sin 30^\circ \) and \( \cos 30^\circ \).
• Calculated \( \sin^4 30^\circ \) and \( \cos^4 30^\circ \).
• Summed the fourth powers: \( \frac{1}{16} + \frac{9}{16} = \frac{10}{16} = \frac{5}{8} \).
• Recognized the second part as a sum/difference identity or used complementary angles.
• Evaluated \( \sin 25^\circ \cos 65^\circ + \sin 65^\circ \cos 25^\circ \) as \( \sin(25^\circ + 65^\circ) = \sin 90^\circ = 1 \).
• Combined the results: \( \frac{5}{8} - 1 = -\frac{3}{8} \).

Revision Table: Key Trigonometric Values & Identities

Important Identities Used:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Complementary Angle Identity: \( \cos(90^\circ - \theta) = \sin \theta \) and \( \sin(90^\circ - \theta) = \cos \theta \)
• Sum Formula for Sine: \( \sin(A+B) = \sin A \cos B + \cos A \sin B \)

Additional Information: Understanding Higher Powers of Sine and Cosine

Calculating higher powers like \( \sin^4 \theta \) and \( \cos^4 \theta \) simply means raising the value of \( \sin \theta \) or \( \cos \theta \) to the power of 4. For example, \( \sin^4 30^\circ = (\sin 30^\circ)^4 \). There are also reduction formulas for higher powers, but for standard angles like \( 30^\circ \) where the basic values are known, direct calculation is straightforward.

The sum formula for sine, \( \sin(A+B) = \sin A \cos B + \cos A \sin B \), is a fundamental identity derived from the unit circle or geometric proofs. It allows us to find the sine of a sum of two angles if we know the sines and cosines of the individual angles. In this problem, recognizing the pattern \( \sin A \cos B + \cos A \sin B \) with \( A=25^\circ \) and \( B=65^\circ \) was key to simplifying the second part of the expression quickly by calculating \( \sin(25^\circ+65^\circ) = \sin 90^\circ \).

Complementary angle identities are useful when dealing with angles that add up to \( 90^\circ \). They show the relationship between trigonometric functions of an angle and its complement. For instance, \( \cos 65^\circ = \sin 25^\circ \) because \( 65^\circ + 25^\circ = 90^\circ \).