Simplify Trigonometric Expression: No Calculator Needed!

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Hey guys! Today, we're diving into a fun little trigonometric problem that might seem tricky at first glance. But don't worry, we'll break it down step by step using trigonometric identities. No calculators allowed – this is all about understanding the concepts! So, let's get started and simplify the following expression:

(cos⁡31∘)(cos⁡59∘)−(sin⁡31∘)(sin⁡59∘){\left(\cos 31^{\circ}\right)\left(\cos 59^{\circ}\right)-\left(\sin 31^{\circ}\right)\left(\sin 59^{\circ}\right)}

Understanding the Problem

Before we jump into solving, let's understand what we're dealing with. The expression involves cosine and sine functions of angles 31° and 59°. Our goal is to simplify this expression using trigonometric identities, specifically the cosine addition formula. This formula will allow us to combine these terms into a single, simpler expression. Remember, the key here is to recognize patterns and apply the correct identities. So, keep your thinking caps on, and let's transform this expression!

Identifying the Correct Trigonometric Identity

Okay, so the expression we're tackling looks a lot like one of our favorite trigonometric identities. Specifically, it resembles the cosine addition formula. The cosine addition formula states:

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B{\cos(A + B) = \cos A \cos B - \sin A \sin B}

Now, if we compare this identity with the given expression:

(cos⁡31∘)(cos⁡59∘)−(sin⁡31∘)(sin⁡59∘){\left(\cos 31^{\circ}\right)\left(\cos 59^{\circ}\right)-\left(\sin 31^{\circ}\right)\left(\sin 59^{\circ}\right)}

We can see that:

  • A = 31°
  • B = 59°

So, our expression is essentially the right-hand side of the cosine addition formula. This is great news because it means we can simplify it into the left-hand side, which is just ${\cos(A + B)}$. This recognition is crucial, and it makes the simplification process much easier. By identifying the correct identity, we've already done a significant part of the work!

Applying the Trigonometric Identity

Alright, now that we've identified the correct trigonometric identity, let's apply it to our expression. We know that:

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B{\cos(A + B) = \cos A \cos B - \sin A \sin B}

And we've determined that A = 31° and B = 59°. So, we can rewrite our expression as:

cos⁡(31∘+59∘){\cos(31^{\circ} + 59^{\circ})}

Now, we just need to add the angles:

cos⁡(31∘+59∘)=cos⁡(90∘){\cos(31^{\circ} + 59^{\circ}) = \cos(90^{\circ})}

So, now our expression is simplified to ${\cos(90^{\circ})}$. This is a significant step because we've reduced the original complex expression into a single cosine function. Next, we'll evaluate this cosine function to get our final answer.

Evaluating the Simplified Expression

Okay, so we've simplified our expression down to ${\cos(90^{\circ})}$. Now, we need to evaluate this. Think back to your unit circle or trigonometric values for special angles. What is the cosine of 90 degrees?

cos⁡(90∘)=0{\cos(90^{\circ}) = 0}

That's right! The cosine of 90 degrees is 0. So, our final simplified answer is 0. This means that the original expression:

(cos⁡31∘)(cos⁡59∘)−(sin⁡31∘)(sin⁡59∘){\left(\cos 31^{\circ}\right)\left(\cos 59^{\circ}\right)-\left(\sin 31^{\circ}\right)\left(\sin 59^{\circ}\right)}

simplifies to 0. How cool is that?

Final Answer

So, after applying the cosine addition formula and evaluating the simplified expression, we arrive at our final answer:

(cos⁡31∘)(cos⁡59∘)−(sin⁡31∘)(sin⁡59∘)=0{\left(\cos 31^{\circ}\right)\left(\cos 59^{\circ}\right)-\left(\sin 31^{\circ}\right)\left(\sin 59^{\circ}\right) = 0}

Therefore, the simplified value of the given expression is 0. This exercise demonstrates the power of trigonometric identities in simplifying complex expressions. By recognizing the pattern and applying the appropriate identity, we were able to solve this problem without using a calculator. Great job, guys!

Key Takeaways

  • Recognize Trigonometric Identities: The ability to recognize trigonometric identities is crucial for simplifying expressions. In this case, identifying the cosine addition formula was the key to solving the problem.
  • Apply Identities Correctly: Once you've identified the correct identity, make sure to apply it correctly. Pay attention to the signs and angles involved.
  • Simplify Step by Step: Break down the problem into smaller, manageable steps. This makes the simplification process easier to follow and reduces the chances of making mistakes.
  • Evaluate Special Angles: Knowing the trigonometric values for special angles (such as 0°, 30°, 45°, 60°, and 90°) is essential for evaluating simplified expressions.

Practice Problems

To solidify your understanding of trigonometric identities, try simplifying the following expressions:

  1. sin⁡40∘cos⁡20∘+cos⁡40∘sin⁡20∘{\sin 40^{\circ} \cos 20^{\circ} + \cos 40^{\circ} \sin 20^{\circ}}

  2. cos⁡75∘cos⁡15∘+sin⁡75∘sin⁡15∘{\cos 75^{\circ} \cos 15^{\circ} + \sin 75^{\circ} \sin 15^{\circ}}

  3. sin⁡(x+y)−sin⁡(x−y){\sin(x + y) - \sin(x - y)}

Work through these problems, and don't hesitate to review the trigonometric identities if you get stuck. Practice makes perfect!

Conclusion

Alright, that wraps up our journey of simplifying trigonometric expressions without a calculator! Remember, the key is to recognize those identities and apply them correctly. With a bit of practice, you'll be simplifying trigonometric expressions like a pro in no time. Keep up the great work, and I'll catch you in the next math adventure!

Keep practicing and have fun with trigonometry!