Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a sum or difference of trigonometric functions or values. $$2 \cos 85^{\circ} \sin 140^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 225^{\circ} + \sin 55^{\circ}\)

Step 1: Identify the Trigonometric Identity

The given expression is in the form of \(2 \cos A \sin B\). This matches the formula for the sum-to-product identity: \(2 \cos A \sin B = \sin(A + B) - \sin(A - B)\).

Step 2: Substitute the Values

Substitute \(A = 85^{\circ}\) and \(B = 140^{\circ}\) into the identity formula. This gives \(\sin(85^{\circ} + 140^{\circ}) - \sin(85^{\circ} - 140^{\circ})\).

Step 3: Simplify the Angles

Calculate the angles: \(85^{\circ} + 140^{\circ} = 225^{\circ}\) and \(85^{\circ} - 140^{\circ} = -55^{\circ}\).

Step 4: Rewrite the Expression

The expression is now \(\sin 225^{\circ} - \sin(-55^{\circ})\).

Step 5: Use Properties of Sine Function

Remember that \(\sin(-\theta) = -\sin(\theta)\), so \(\sin(-55^{\circ}) = -\sin(55^{\circ})\). Substitute it back into the expression to get \(\sin 225^{\circ} + \sin 55^{\circ}\).

Step 6: Final Expression

Therefore, the original expression \(2 \cos 85^{\circ} \sin 140^{\circ}\) can be written as \(\sin 225^{\circ} + \sin 55^{\circ}\).

Key Concepts

sum-to-product identities

Sum-to-product identities are incredibly useful in trigonometry for simplifying expressions. When you have an expression like \(2 \cos A \sin B\), it can transform into a simpler form using these identities. The formula for this specific expression is:

• \( 2 \cos A \sin B = \sin(A + B) - \sin(A - B) \)

This identity is part of a broader set that arises from combining sum and difference formulas of sine and cosine functions. By recognizing that the exercise matches the form \(2 \cos A \sin B\), we can quickly apply the identity.
In practical terms, this identity helps convert the product of sine and cosine into a sum or difference of sine functions. This is particularly useful when dealing with integrals or solving trigonometric equations.

sine function properties

The sine function has several important properties that can simplify trigonometric expressions. Understanding these helps us navigate through exercises efficiently.

• Odd Function: The sine function is an odd function, which means \(\sin(-\theta) = -\sin(\theta)\).
• Range and Periodicity: The range of the sine function is \, [-1, 1] \, and its period is \(360^\circ\) or \(2\pi\; \text{radians}\). Thus, after \(360^\circ\), the sine values repeat.\

Applying these properties, we can transform expressions like \(\sin(-55^{\circ})\) into positive angle equivalents, knowing \(\sin(-55^{\circ})\) simplifies to \(-\sin(55^{\circ})\). This makes it easier to manage computations, as seen in the step where \(\sin(225^{\circ}) + \sin(55^{\circ})\) becomes the final expression.

angle transformation

Angle transformation in trigonometry allows us to convert angles into more manageable forms or to utilize identities effectively. Consider the original expression where you have to calculate \(85^{\circ} + 140^{\circ}\) and \(85^{\circ} - 140^{\circ}\):

• Sum of Angles: Here, the sum \(225^{\circ}\) doesn't require transformation as it lies within the standard range. \
• Difference of Angles: The difference \(-55^{\circ}\) needs transformation by understanding sine properties, resulting in \(-\sin(55^{\circ})\).\

Using these transformations, trigonometric problems become more approachable, turning what seems like complex calculations into systematic steps. Angle transformations particularly enhance problem-solving efficiency by aligning with easily recognizable identities.