91Ó°ÊÓ

The sine double angle identity is a helpful tool in trigonometry that simplifies expressions involving sine and cosine. This identity is expressed as:

• \( \sin 2A = 2 \sin A \cos A \)

It essentially states that the sine of twice an angle is equal to twice the product of the sine and cosine of that angle. This can simplify computations and evaluations of sine and cosine with compound angles.
In the given exercise, the problem involves the expression \( \sin 15^{\circ} \cos 15^{\circ} \). By recognizing this as half of our double angle formula, we use the identity to rewrite the expression in terms of a single angle, \( \sin 30^{\circ} \), making it simpler to evaluate.

Exact value evaluation involves calculating the trigonometric function value without a calculator, using known values and identities. Common exact values come from angles like \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\).
For \( \sin 30^{\circ} \), which is derived from the identity mentioned earlier, we know from trigonometric tables or the unit circle that:

• \( \sin 30^{\circ} = \frac{1}{2} \)

This step is crucial in our problem because it allows us to substitute back accurately to reach our final, simplified expression. Evaluating these exact values ensures precision and builds confidence in solving more complex problems. The final simplified expression becomes \( \frac{1}{4} \) once the substitution and multiplication are complete.

Trigonometric simplification involves reducing expressions into simpler forms, often using identities. In this task, the goal was to simplify \( \sin 15^{\circ} \cos 15^{\circ} \).
The process involves:

• Recognizing which identity can apply, like the sine double angle identity in our example.
• Substituting given angles into the identity to simplify terms, in this case, using \( 2A = 30^{\circ} \).
• Finally, calculating the precise trigonometric values, like \( \sin 30^{\circ} \), and evaluating them to further simplify.

This step-by-step simplification helps manage different elements of an equation. It improves understanding of how different trigonometric functions relate and also enhances problem-solving skills required for more advanced mathematics.