Trigonometric Expression Evaluation: A Step-by-Step Guide

Hey guys! Today, we're diving into a trigonometric problem that might seem a bit daunting at first, but don't worry, we'll break it down together. We need to evaluate the expression:

$$5\sin(\alpha + 2\pi) + 11\cos\left(-\frac{\pi}{2} + \alpha\right)$$

Given that $\sin \alpha = 0.8$. Sounds fun, right? Let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what's being asked. We're given a trigonometric expression involving sine and cosine functions with angle transformations. Our mission, should we choose to accept it (and we do!), is to simplify this expression using trigonometric identities and then plug in the given value of $\sin \alpha$ to find the final answer.

Step 1: Simplifying the Sine Term

Let's tackle the first term: $5\sin(\alpha + 2\pi)$. The key here is to remember that the sine function has a period of $2\pi$. This means that adding $2\pi$ to the angle doesn't change the sine value. In other words:

$$\sin(\alpha + 2\pi) = \sin(\alpha)$$

So, our first term simplifies to:

$$5\sin(\alpha + 2\pi) = 5\sin(\alpha)$$

This is a crucial simplification, so make sure you're comfortable with this periodic property of sine.

Step 2: Simplifying the Cosine Term

Now let's look at the second term: $11\cos\left(-\frac{\pi}{2} + \alpha\right)$. This one requires a bit more finesse. We'll use the cosine cofunction identity, which states:

$$\cos\left(-\frac{\pi}{2} + \alpha\right) = \sin(\alpha)$$

This identity might look intimidating, but it's a powerful tool. Think of it this way: cosine of an angle is the same as the sine of its complement (the angle that adds up to $\frac{\pi}{2}$). In our case, the complement of $(-\frac{\pi}{2} + \alpha)$ is $\alpha$.

So, our second term simplifies beautifully to:

$$11\cos\left(-\frac{\pi}{2} + \alpha\right) = 11\sin(\alpha)$$

Mastering cofunction identities will significantly boost your trig skills.

Step 3: Putting It All Together

Now that we've simplified both terms, let's rewrite the entire expression:

$$5\sin(\alpha + 2\pi) + 11\cos\left(-\frac{\pi}{2} + \alpha\right) = 5\sin(\alpha) + 11\sin(\alpha)$$

See how much simpler it looks now? We can combine these terms since they both involve $\sin(\alpha)$:

$$5\sin(\alpha) + 11\sin(\alpha) = 16\sin(\alpha)$$

This is the power of simplification! We've reduced a complex expression to a single term.

Step 4: Plugging in the Value of sin α

We're given that $\sin \alpha = 0.8$. Now it's the home stretch! We simply substitute this value into our simplified expression:

$$16\sin(\alpha) = 16 \times 0.8$$

Performing the multiplication, we get:

$$16 \times 0.8 = 12.8$$

And there you have it! We've successfully evaluated the expression.

Final Answer

The value of the expression $5\sin(\alpha + 2\pi) + 11\cos\left(-\frac{\pi}{2} + \alpha\right)$, given that $\sin \alpha = 0.8$, is 12.8.

Key Takeaways

• Periodic Properties: Remember that $\sin(\alpha + 2\pi) = \sin(\alpha)$. This is crucial for simplifying trigonometric expressions.
• Cofunction Identities: Mastering identities like $\cos\left(-\frac{\pi}{2} + \alpha\right) = \sin(\alpha)$ is essential.
• Simplify First: Always try to simplify the expression before plugging in values. It makes the calculations much easier.
• Practice Makes Perfect: The more you practice these types of problems, the more comfortable you'll become with trigonometric identities and simplifications.

Let's Dig Deeper: Understanding Trigonometric Identities

Trigonometric identities are like the secret sauce of trigonometry. They allow us to rewrite trigonometric expressions in different forms, making them easier to work with. Understanding these identities isn't just about memorizing formulas; it's about grasping the relationships between different trigonometric functions and angles.

The Unit Circle Connection

Many trigonometric identities can be visualized and understood using the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. For any angle $\theta$, the point where the terminal side of the angle intersects the unit circle has coordinates $(\cos \theta, \sin \theta)$. This visual representation helps to see how sine and cosine values change as the angle changes, which is fundamental to understanding their periodic behavior.

For example, consider the identity $\sin(\alpha + 2\pi) = \sin(\alpha)$. On the unit circle, adding $2\pi$ to an angle simply means going around the circle once, ending up at the same point. Therefore, the sine value (the y-coordinate) remains the same.

Common Trigonometric Identities

Let's take a quick look at some common trigonometric identities that are useful for simplifying expressions and solving equations:

• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. This is perhaps the most fundamental trigonometric identity, derived directly from the Pythagorean theorem.
• Angle Sum and Difference Identities: These identities allow you to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles.
  • $\sin(A + B) = \sin A \cos B + \cos A \sin B$
  • $\sin(A - B) = \sin A \cos B - \cos A \sin B$
  • $\cos(A + B) = \cos A \cos B - \sin A \sin B$
  • $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• Double-Angle Identities: These identities are derived from the angle sum identities and express trigonometric functions of twice an angle in terms of trigonometric functions of the angle itself.
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$
• Cofunction Identities: We used one of these in our problem! These identities relate trigonometric functions of complementary angles (angles that add up to $\frac{\pi}{2}$). For example:
  • $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$
  • $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$

Tips for Using Trigonometric Identities

• Memorization is Key (But Understanding is More Important): While memorizing identities is helpful, understanding why they work is even more important. This will allow you to apply them more flexibly and effectively.
• Look for Patterns: Practice recognizing common patterns and structures in trigonometric expressions. This will help you identify which identities to use.
• Don't Be Afraid to Experiment: Sometimes, the best way to simplify an expression is to try different identities and see what works. There's often more than one way to solve a problem!
• Use the Unit Circle: The unit circle is a powerful tool for visualizing trigonometric relationships and understanding identities.

Practice Problems

To solidify your understanding of trigonometric identities and expression evaluation, try working through these practice problems:

1. Simplify the expression $\cos(\alpha + \pi)$ given that $\cos(\alpha) = 0.6$.
2. Evaluate $2\sin(x)\cos(x)$ if $\sin(x) = \frac{1}{3}$ and $x$ is in the first quadrant.
3. Prove the identity $\frac{\sin(2x)}{1 + \cos(2x)} = \tan(x)$.

Remember, the key to mastering trigonometry is consistent practice and a solid understanding of the fundamental concepts.

Conclusion

Evaluating trigonometric expressions might seem tricky at first, but by breaking down the problem into smaller steps and using trigonometric identities, we can simplify even the most complex expressions. Guys, keep practicing, and you'll become trig wizards in no time! Remember the key takeaways: periodic properties, cofunction identities, simplification, and practice. You've got this!