Problem 76
Question
$$ \frac{1}{2} \sin 3 x+\frac{\sqrt{3}}{2} \cos 3 x=\sin 5 x $$
Step-by-Step Solution
Verified Answer
The equation is true when \( x = 15^{\degree} \). This is achieved by recognizing the structure as part of the sine sum / difference formula and applying that knowledge to transform the original equation.
1Step 1: Recognize the identity structure
The left hand side of the equation \( \frac{1}{2} \sin 3 x+\frac{\sqrt{3}}{2} \cos 3 x\) resembles the structure of the sine sum or difference formula. The sine sum formula is \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). This will lead us to recognize that \(\frac{1}{2}\) can be written as \( \cos 30^{\degree}\) and \( \frac{\sqrt{3}}{2}\) can be recognized as \( \sin 30^{\degree}\).
2Step 2: Apply the sine sum formula
Plugging \( a = 3x \) and \( b = 30^{\degree} \) into the sine sum formula, we get \( \sin(a + b) = \cos 30^{\degree} \sin 3 x + \sin 30^{\degree} \cos 3 x\). This is equivalent to the left side of the original equation.
3Step 3: Simplification
Recognize that \( a + b = 3x + 30^{\degree} = 90^{\degree} + x \), which equals \( sin (90^{\degree} + x) \) or its equivalent form \( cos x \). We can transform \( cos x \) to \( sin(90^{\degree} - x) \) or \( sin(5x) \) since \( 5x = 90^{\degree} - x \) when \( x = 15^{\degree} \). Previewing that our simplification is correct and we successfully prove the equation.
Key Concepts
Sine Sum FormulaAngle ConversionSimplification Techniques
Sine Sum Formula
The sine sum formula is an essential trigonometric identity that helps to simplify expressions of sums of angles. When you're faced with an expression like \( \sin(a + b) \), the sine sum formula comes into play. It states:
In this exercise, we recognize a similar structure with the expression \( \frac{1}{2} \sin 3x + \frac{\sqrt{3}}{2} \cos 3x \). We see that \( \frac{1}{2} \) and \( \frac{\sqrt{3}}{2} \) resemble the cosine and sine values of \( 30^{\degree} \), respectively. This leads us to use the sine sum formula to transform the expression into a single sine function of the sum of angles, facilitating the simplification process.
- \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
In this exercise, we recognize a similar structure with the expression \( \frac{1}{2} \sin 3x + \frac{\sqrt{3}}{2} \cos 3x \). We see that \( \frac{1}{2} \) and \( \frac{\sqrt{3}}{2} \) resemble the cosine and sine values of \( 30^{\degree} \), respectively. This leads us to use the sine sum formula to transform the expression into a single sine function of the sum of angles, facilitating the simplification process.
Angle Conversion
Angle conversion is crucial when working with trigonometric identities since it bridges different angle systems and simplifies calculations. Here, understanding that angles in trigonometry can be represented in degrees or radians is essential.
Converting between systems or recognizing familiar angle sine and cosine values helps in identifying applicable identities.
In our problem, we notice that the constants \( \frac{1}{2} \text{ and } \frac{\sqrt{3}}{2} \) are key trigonometric ratios for \( 30^{\degree} \) and \( 60^{\degree} \). This recognition is vital for correctly applying the sine sum formula and swapping parts of the expression into tractable forms using known angle conversions.
Converting between systems or recognizing familiar angle sine and cosine values helps in identifying applicable identities.
In our problem, we notice that the constants \( \frac{1}{2} \text{ and } \frac{\sqrt{3}}{2} \) are key trigonometric ratios for \( 30^{\degree} \) and \( 60^{\degree} \). This recognition is vital for correctly applying the sine sum formula and swapping parts of the expression into tractable forms using known angle conversions.
Simplification Techniques
Simplification is the process of rewriting expressions in more manageable forms, especially when dealing with trigonometric identities. In this particular example, simplification involves transforming the expression into more familiar trigonometric terms.
- After applying the sine sum formula, we simplify \( \sin(3x + 30^{\circ}) \) to reach other equivalent forms.
- Understanding that \( 3x + 30^{\circ} \) can be rewritten as \( 90^{\circ} + x \), leading to recognized forms like \( \sin(90^{\circ} + x) \) and equivalent identity transformations.
- This further links to concepts where \( \sin(90^{\circ} + x) = \cos x \), due to complementary angle identities. Thus forming a connection between the given and the required expression of \( \sin 5x \) when \( 5x = 90^{\circ} - x \) for \( x = 15^{\circ} \).