Problem 103 - Precalculus Mathematics for Calculus

Question

If \(A, B,\) and \(C\) are the angles in a triangle, show that $$\sin 2 A+\sin 2 B+\sin 2 C=4 \sin A \sin B \sin C$$

Step-by-Step Solution

Answer

The equation is true: \( \sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C \).

Step 1: Recall Angle Sum in a Triangle

In any triangle, the sum of the angles is given by: \( A + B + C = 180^\circ \). This relationship is crucial as it will allow us to express one angle in terms of the others.

Step 2: Use the Sine of Angle Sum Formula

Using the angle sum identity for sine, we express \( \sin 2C \) as: \( \sin(2 \times (180^\circ - A - B)) = \sin(360^\circ - 2A - 2B) = - \sin(2A + 2B) \).

Step 3: Express \( \sin 2A + \sin 2B \)

Apply the sum-to-product identity, \( \sin 2A + \sin 2B = 2 \sin(A + B) \cos(A - B) \).

Step 4: Substitute for \( \sin 2C \)

From Step 2, substitute \( \sin 2C = - \sin(2A + 2B) = - 2 \sin(A + B) \cos(A + B) \) into the equation.

Step 5: Combine Expressions

Combine the expressions from Steps 3 and 4 to get: \( 2 \sin(A + B) \cos(A - B) - 2 \sin(A + B) \cos(A + B) \).

Step 6: Factor Out Common Terms

Factor \( 2 \sin(A + B) \) from the expression: \( 2 \sin(A + B)(\cos(A - B) - \cos(A + B)) \).

Step 7: Simplify Using Trigonometric Identity

Use the identity \( \cos A - \cos B = -2 \sin((A + B)/2) \sin((A - B)/2) \) to simplify the expression to \( -4 \sin(A + B) \sin A \sin B \).

Step 8: Interpretation Using Triangle Angles

Recognize that \( A + B = 180^\circ - C \), thus \( \sin(A + B) = \sin C \). This simplifies the expression to \( -4 \sin C \sin A \sin B \). Since both sides are essentially equal and positive, equate to \( 4 \sin A \sin B \sin C \).

Key Concepts

Angle Sum in a Triangle

In any triangle, the angles add up to a special total: 180 degrees. This is a fundamental property of triangles that helps us understand and solve problems involving angles.
For instance, if you know two angles, you can always find the third by subtracting the sum of the known angles from 180 degrees. This property becomes particularly useful when working with trigonometric identities, as it allows us to express one angle in terms of the other two.
Here, if the triangle has angles labeled as \( A, B, \) and \( C \), the relationship is expressed as:
\[A + B + C = 180^\circ\]Understanding this concept is crucial, as it allows us to simplify the expressions during trigonometric calculations. When dealing with identities involving these angles, knowing that they collectively sum to 180 degrees enables us to manipulate and transform the expressions more effectively.

Sine Angle Sum Formula

The sine angle sum formula is a powerful tool in trigonometry. It tells us how to find the sine of the sum of two angles. This formula is especially useful when you want to break down complex angle expressions into more manageable parts.
For the sum of two angles \(A\) and \(B\), the formula is:
\[\sin(A + B) = \sin A \cos B + \cos A \sin B\] In our exercise, this formula helps us transform \(\sin 2C\). Since the angle sum in a triangle is 180 degrees, we rewrite \(C\) in terms of \(A\) and \(B\). When applying the formula, remember to account for the angle properties of a triangle.
This ensures we're expressing the sines in terms of angles that still fit neatly into our known sums or differences, which ties directly back to the critical relationships between the triangle's angles. Incorporating the sine angle sum formula makes it simpler to work through and solve such problems strategically.

Sum-to-Product Identities

Sum-to-product identities are special trigonometric formulas that allow you to convert sums or differences of sines and cosines into products. These identities ease certain calculations by enabling you to change the format of the expression.

• For two angles \(A\) and \(B\), the identity for the sum of sines is: \[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
• Similarly, for their difference: \[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]

These identities help transform the expression into a simpler or more usable form.
In our specific problem, the sum-to-product identity transforms \(\sin 2A + \sin 2B\) into a more manageable expression involving cosine. With this transformation, merging it with other trigonometric identities is more straightforward. Ensuring you understand how to apply these identities well can greatly simplify trigonometric calculations, making the process not only faster but also less prone to errors.