Sin 165°: Finding Exact Values With Sum/Difference Formula

Have you ever wondered how to find the exact value of a trigonometric function for an angle that isn't a standard one like 30°, 45°, or 60°? In this comprehensive guide, we'll dive deep into the world of trigonometry, specifically focusing on how to calculate the exact value of ${\sin 165^{\circ}}$ using the sum and difference formulas. We'll break down each step, making it easy to follow along, and by the end, you'll not only know the answer but also understand the underlying principles behind it. Let’s embark on this mathematical journey together!

Understanding the Sum and Difference Formulas

Before we tackle ${\sin 165^{\circ}}$, it's essential to grasp the fundamental concept of sum and difference formulas in trigonometry. These formulas allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. For sine, the sum and difference formulas are:

• ${\sin(A + B) = \sin A \cos B + \cos A \sin B}$
• ${\sin(A - B) = \sin A \cos B - \cos A \sin B}$

These formulas are the cornerstones of our calculation. The key idea is to break down 165° into a sum or difference of angles whose sine and cosine values we already know. Common angles to consider are 30°, 45°, 60°, and their multiples, as these correspond to special triangles (30-60-90 and 45-45-90 triangles) with well-known trigonometric ratios. By mastering these formulas, you unlock a powerful tool for solving a wide array of trigonometric problems, extending beyond just finding sine values. Understanding these principles lays a solid foundation for more advanced concepts in trigonometry and calculus.

Expressing 165° as a Sum of Special Angles

The crucial step in finding the exact value of ${\sin 165^{\circ}}$ is to express 165° as a sum or difference of angles with known trigonometric values. Think about which special angles add up to 165°. One convenient way is to express 165° as the sum of 120° and 45° because we know the exact sine and cosine values for both of these angles. Another option is to consider 165° as the sum of 135° and 30°. Both decompositions work, but for this explanation, let’s use 165° = 120° + 45°. This approach sets the stage for applying the sine sum formula effectively. The choice of angles can sometimes impact the complexity of the calculation, but understanding the flexibility in expressing the target angle is key. By practicing with different angle combinations, you can develop an intuition for which combinations will lead to the most straightforward solutions. This skill is not only valuable for this particular problem but also for a broader range of trigonometric challenges.

Applying the Sine Sum Formula

Now that we've expressed 165° as 120° + 45°, we can apply the sine sum formula:

${\sin(A + B) = \sin A \cos B + \cos A \sin B}$

In our case, A = 120° and B = 45°. So, we have:

${\sin 165^{\circ} = \sin(120^{\circ} + 45^{\circ}) = \sin 120^{\circ} \cos 45^{\circ} + \cos 120^{\circ} \sin 45^{\circ}}$

This step transforms the original problem into evaluating the trigonometric functions of familiar angles. The sine sum formula acts as a bridge, connecting the unknown value of ${\sin 165^{\circ}}$ to the known values of sines and cosines of 120° and 45°. The accuracy of this step is paramount, as any error in applying the formula will propagate through the rest of the calculation. It's helpful to write out the formula explicitly before substituting the values, ensuring that each term is correctly placed. This meticulous approach minimizes the risk of mistakes and sets the stage for a successful final answer. By understanding the mechanics of the sine sum formula, you gain a deeper appreciation for its utility in simplifying complex trigonometric expressions.

Evaluating the Trigonometric Functions

Next, we need to evaluate the trigonometric functions for 120° and 45°. These are special angles, and their sine and cosine values are well-known:

• ${\sin 120^{\circ} = \frac{\sqrt{3}}{2}}$
• ${\cos 120^{\circ} = -\frac{1}{2}}$
• ${\sin 45^{\circ} = \frac{\sqrt{2}}{2}}$
• ${\cos 45^{\circ} = \frac{\sqrt{2}}{2}}$

These values are derived from the unit circle or special right triangles (30-60-90 and 45-45-90 triangles). Memorizing these values is incredibly helpful, but understanding how they are derived is even more important. Knowing the geometry behind these values allows you to reconstruct them even if you forget them, and it reinforces your understanding of the relationship between angles and trigonometric ratios. This step is a critical link in the chain, and accurate values are essential for obtaining the correct final result. It's always a good idea to double-check these values, especially in high-stakes situations like exams.

Substituting the Values

Now, substitute these values back into our equation:

${\sin 165^{\circ} = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)}$

This step is a straightforward substitution, but it's crucial to maintain accuracy. Double-check that each value is placed correctly and that the signs are consistent. The equation now contains only numerical values, making it possible to simplify and arrive at the final answer. This process of substitution is a common technique in mathematics, transforming an abstract equation into a concrete expression that can be directly evaluated. The clarity and organization of this step significantly impact the likelihood of arriving at the correct result. A clean substitution makes subsequent simplifications much easier to manage.

Simplifying the Expression

Let's simplify the expression:

${\sin 165^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}$

Combine the terms:

${\sin 165^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}}$

This simplification involves multiplying fractions and combining like terms. The key here is to be meticulous with the arithmetic. Each step must be performed correctly to maintain the integrity of the expression. Recognizing the common denominator allows for a straightforward combination of the terms, leading to a more compact form. This simplification step demonstrates the power of algebraic manipulation in making complex expressions more manageable. The goal is to arrive at the simplest form of the answer, which not only makes the result easier to interpret but also reduces the chance of errors in further calculations.

Matching the Given Form and Finding A and B

We are given that:

${\sin 165^{\circ} = \frac{\sqrt{A}(\sqrt{B}-1)}{4}}$

We have found that:

${\sin 165^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}}$

To match the given form, we can rewrite the expression as:

${\sin 165^{\circ} = \frac{\sqrt{2}(\sqrt{3}-1)}{4}}$

Comparing the two forms, we can identify A and B. This final step requires a bit of algebraic insight. The goal is to rearrange the expression we derived to match the format provided in the problem statement. This often involves factoring out a common factor or manipulating the terms in a way that reveals the desired structure. In this case, factoring out ${\sqrt{2}}$ is the key to making the match. This skill of pattern recognition and algebraic manipulation is crucial for solving many types of mathematical problems. Once the expression is in the correct format, identifying the values of A and B is a straightforward comparison.

Thus:

• A = 2
• B = 3

Conclusion

Congratulations! You've successfully navigated the process of finding the exact value of ${\sin 165^{\circ}}$ using the sum formula. By breaking down the problem into manageable steps, applying the appropriate trigonometric identities, and carefully simplifying the expressions, we arrived at the solution. Remember, the key to mastering trigonometry lies in understanding the underlying concepts and practicing regularly. We found that A = 2 and B = 3 when ${\sin 165^{\circ}}$ is expressed in the form ${\frac{\sqrt{A}(\sqrt{B}-1)}{4}}$. This journey demonstrates the power of trigonometric identities in simplifying complex calculations and revealing elegant solutions. The techniques we've explored here are applicable to a wide range of trigonometric problems, making this a valuable skill to develop. Keep practicing, and you'll become more confident and proficient in your trigonometric endeavors.

For further exploration and practice with trigonometric identities, you might find valuable resources on websites like Khan Academy's Trigonometry Section.