Evaluate Sine Expression Using Sum Identity: Step-by-Step

In this article, we'll break down how to evaluate the expression $\sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ}$ using the sum identity for sine. This is a common type of problem in trigonometry, and understanding how to approach it will help you tackle similar questions with confidence. We will cover the sine sum identity, how to apply it to the given expression, and arrive at the final answer. Let's dive in!

Understanding the Sine Sum Identity

The sine sum identity is a fundamental trigonometric identity that allows us to express the sine of the sum of two angles in terms of the sines and cosines of the individual angles. This identity is expressed as:

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

Where A and B represent angles. This identity is crucial for simplifying expressions and solving trigonometric equations. Mastering this identity is essential for anyone studying trigonometry, as it appears in various applications, including physics, engineering, and other fields that rely on wave phenomena and oscillations. The sine sum identity is not just a standalone formula; it’s a cornerstone in the world of trigonometry, connecting seemingly disparate trigonometric functions in a cohesive and predictable way. By understanding how to use this identity, you’re unlocking a powerful tool that simplifies complex trigonometric problems into manageable steps.

The beauty of the sine sum identity lies in its ability to transform a single trigonometric function of a sum of angles into an expression involving individual angles. This transformation is invaluable when you need to calculate the sine of an angle that isn't a standard value, but can be expressed as the sum of two standard angles (like 30°, 45°, 60°, etc.). In practical terms, the sine sum identity enables us to break down a complex problem into simpler components. For example, if you need to find the sine of 75°, you can express 75° as the sum of 45° and 30°, and then apply the identity using the known sine and cosine values for these standard angles. This approach not only simplifies calculations but also provides a deeper understanding of the relationships between different angles and their trigonometric functions.

In more advanced contexts, the sine sum identity forms the basis for deriving other important trigonometric formulas, such as the double-angle and half-angle formulas. These formulas, in turn, are used in a wide range of mathematical and scientific applications. Furthermore, the sine sum identity is a key element in the study of Fourier analysis, which is used to decompose complex waveforms into simpler sine and cosine waves. This has profound implications in fields such as signal processing, acoustics, and image analysis. Thus, the sine sum identity isn't just a formula to memorize; it’s a gateway to a deeper understanding of trigonometry and its applications in the real world. By grasping this concept firmly, you’ll be well-equipped to tackle a wide array of problems and appreciate the interconnectedness of mathematical ideas.

Applying the Sine Sum Identity to the Expression

Now, let's apply this identity to our expression: $\sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ}$. Notice how this expression closely resembles the right side of the sine sum identity: $\sin A \cos B + \cos A \sin B$. We can see a clear correspondence between the terms.

To effectively use the sine sum identity, we need to identify the values of A and B in our given expression. By comparing $\sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ}$ with the identity $\sin A \cos B + \cos A \sin B$, we can make the following identifications:

• A = 28°
• B = 17°

Now that we have identified A and B, we can substitute these values into the sine sum identity. This substitution will allow us to rewrite the given expression in a simplified form. The sine sum identity, $\sin(A + B) = \sin A \cos B + \cos A \sin B$, is the key to this simplification process. By recognizing the pattern in the given expression and matching it to the identity, we can transform a complex sum of products into a single sine function, which is much easier to evaluate.

Substituting A = 28° and B = 17° into the identity, we get:

$\qquad \sin(28^{\circ} + 17^{\circ}) = \sin 28^{\circ} \cos 17^{\circ} + \cos 28^{\circ} \sin 17^{\circ}$

This step is crucial because it directly applies the theoretical knowledge of the sine sum identity to the specific problem at hand. It demonstrates the power of pattern recognition in mathematics, where identifying a familiar structure allows us to apply a known formula and simplify the problem. The substitution transforms the original expression into a more manageable form, setting the stage for the final calculation.

Simplifying the Expression

With the values of A and B identified, we can simplify the expression. We've established that:

$\qquad \sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ} = \sin(28^{\circ} + 17^{\circ})$

The next step is to add the angles within the sine function:

$\qquad 28^{\circ} + 17^{\circ} = 45^{\circ}$

Now, our expression simplifies to:

$\qquad \sin(45^{\circ})$

This is a significant simplification because 45° is a standard angle for which we know the exact sine value. By reducing the original expression to the sine of a single, well-known angle, we've made it much easier to evaluate. This step highlights the efficiency of using trigonometric identities to simplify complex expressions into more manageable forms. The ability to reduce a problem to its simplest elements is a hallmark of mathematical problem-solving, and this step exemplifies that process. The simplification not only makes the calculation straightforward but also provides a clear path to the final answer.

Evaluating $\sin(45^{\circ})$

The final step is to evaluate $\sin(45^{\circ})$. The sine of 45 degrees is a commonly known value in trigonometry. It represents the y-coordinate of the point on the unit circle that corresponds to an angle of 45 degrees (or $\frac{\pi}{4}$ radians). Understanding the unit circle and the sine values for standard angles is essential for solving trigonometric problems efficiently. The value of $\sin(45^{\circ})$ is a fundamental piece of trigonometric knowledge, and recalling it quickly can significantly speed up problem-solving.

From our knowledge of trigonometric values, we know that:

$\qquad \sin(45^{\circ}) = \frac{\sqrt{2}}{2}$

Therefore, the value of the original expression $\sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ}$ is $\frac{\sqrt{2}}{2}$. This result is an exact value, meaning it doesn't involve any approximations or rounding. Expressing the answer in exact form is often preferred in mathematics, as it avoids any loss of precision. The final answer, $\frac{\sqrt{2}}{2}$, is a testament to the power of trigonometric identities in simplifying complex expressions and arriving at elegant, precise solutions.

Conclusion

In summary, by recognizing the structure of the expression $\sin 28^{\circ} \cos 17^{\circ} + \sin 17^{\circ} \cos 28^{\circ}$ and applying the sine sum identity, we were able to simplify it to $\sin(45^{\circ})$, which equals $\frac{\sqrt{2}}{2}$. This process demonstrates the power and utility of trigonometric identities in simplifying complex expressions. Remember, the key to mastering trigonometry is understanding and applying these identities effectively.

For further learning on trigonometric identities, you can visit resources like Khan Academy's trigonometry section, which offers comprehensive lessons and practice exercises.