Calculating Sin Α And Cos Α: A Step-by-Step Guide

Hey guys! Today, we're diving into some trigonometry, specifically how to calculate sin α and cos α when we're given some information about double angles. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, making sure everyone understands the process. We'll be tackling two different scenarios, so let's get started. This is useful for anyone trying to understand trigonometric functions or preparing for a math test. Understanding these concepts will provide a solid foundation for more complex trigonometric problems. We will explore how to solve these problems while incorporating useful SEO keywords and strategies.

Understanding the Basics: Trigonometric Identities

Before we jump into the calculations, let's refresh our memory on some key trigonometric identities. These are the tools of the trade, the formulas that will help us solve the problems. The most important one for this is the double-angle formulas. These formulas relate the trigonometric functions of an angle α to those of 2α. Knowing these formulas is absolutely crucial for this type of problem. Let's look at the two main ones we'll need:

• cos 2α = cos²α - sin²α
• sin 2α = 2 sin α cos α

Additionally, the Pythagorean identity is always useful: sin²α + cos²α = 1. Remember these like they're your best friends, because we're going to be using them a lot! Now that we have the groundwork set, let's look at the specifics of what we're going to do. We'll solve these problems using a methodical approach. First, we will identify the given information and what we need to find. Then, we will use the relevant trigonometric identities to get the answer. We will clarify each step, so you can follow it closely. Understanding the concepts covered here, will help a student gain confidence in solving any kind of trigonometric problems.

Case a) Given cos 2α = 1/3 and 2α ∈ (0; π/2)

Okay, guys, let's start with the first part of the problem. We're given that cos 2α = 1/3 and that the angle 2α lies in the interval (0, π/2). This means that 2α is an angle in the first quadrant, where both sine and cosine are positive. Our goal is to find sin α and cos α. So how do we start? Well, we know cos 2α and we want to find cos α and sin α. This is where our double-angle formulas come into play. We can use the formula cos 2α = cos²α - sin²α, but it's not directly helpful. Instead, we can use the following identities:

• cos 2α = 2cos²α - 1
• cos 2α = 1 - 2sin²α

We choose one of these because we know cos 2α and we can easily solve for either cos α or sin α. Let's use cos 2α = 2cos²α - 1. We can substitute 1/3 for cos 2α:

1/3 = 2cos²α - 1.

Now, let's solve for cos²α. Add 1 to both sides: 4/3 = 2cos²α. Divide by 2: cos²α = 2/3. Take the square root of both sides: cos α = ±√(2/3). But wait! Remember that 2α is in the first quadrant, which implies that α must also be in the first quadrant, where cosine is positive. So, we take the positive square root: cos α = √(2/3).

Now we have cos α. Great! To find sin α, we can use the Pythagorean identity: sin²α + cos²α = 1. We already know cos α = √(2/3), so we can substitute that in: sin²α + (2/3) = 1. Subtract 2/3 from both sides: sin²α = 1/3. Take the square root of both sides: sin α = ±√(1/3). And, because α is in the first quadrant, sine is positive as well. So, we have: sin α = √(1/3).

Therefore, for case a), we've found that cos α = √(2/3) and sin α = √(1/3). That wasn't so bad, right?

Case b) Given sin 2α = √5/5 and 2α ∈ (π/2; π)

Alright, let's move on to the second part of the problem, where we're given sin 2α = √5/5 and 2α lies in the interval (π/2, π). This means that 2α is an angle in the second quadrant. In the second quadrant, sine is positive and cosine is negative. Our mission remains the same: find sin α and cos α.

This time, we're given sin 2α, so we will utilize the double-angle formula sin 2α = 2 sin α cos α. We can also use cos²α + sin²α = 1. However, to solve the problem, we will first determine the value of cos 2α. This can be done by using the Pythagorean identity: sin²(2α) + cos²(2α) = 1. Substituting the known value of sin 2α, we have:

(√5/5)² + cos²(2α) = 1.

Simplifying this, we get: 1/5 + cos²(2α) = 1. Subtracting 1/5 from both sides gives us cos²(2α) = 4/5. Taking the square root, we get cos 2α = ±2/√5. Because 2α is in the second quadrant, cosine is negative, therefore cos 2α = -2/√5.

Now we know both sin 2α and cos 2α. To find sin α and cos α, we can use the following identities:

• cos 2α = cos²α - sin²α
• cos 2α = 2cos²α - 1
• cos 2α = 1 - 2sin²α

Using cos 2α = 2cos²α - 1, we have:

-2/√5 = 2cos²α - 1.

Solving for cos²α:

2cos²α = 1 - 2/√5.

cos²α = (1 - 2/√5)/2.

Taking square root cos α = ±√((1 - 2/√5)/2).

Since 2α is in the second quadrant, π/2 < 2α < π, hence π/4 < α < π/2. Hence cos α is positive. Therefore: cos α = √((1 - 2/√5)/2).

Then using sin²α + cos²α = 1, and substituting the value of cos α, we get: sin²α + (1 - 2/√5)/2 = 1.

Solving for sin²α:

sin²α = 1 - (1 - 2/√5)/2.

sin²α = (1 + 2/√5)/2.

Taking square root, we have: sin α = ±√((1 + 2/√5)/2).

Since π/4 < α < π/2, then sin α is also positive. Therefore sin α = √((1 + 2/√5)/2).

So, for case b), we've found that cos α = √((1 - 2/√5)/2) and sin α = √((1 + 2/√5)/2). We did it! We used the double-angle formulas and some good old-fashioned algebraic manipulation to solve for sin α and cos α. Remember to always pay attention to which quadrant the angle falls in, as this affects the signs of sine and cosine.

Conclusion: Mastering Trigonometric Calculations

So, there you have it, guys! We've successfully calculated sin α and cos α in two different scenarios, using the double-angle formulas and the Pythagorean identity. Remember, the key is to break down the problem into smaller steps and to use the right formulas. Practice is key, so don't hesitate to work through more examples. With a bit of practice, you'll become a pro at these types of calculations. Always keep an eye on the quadrants, because the sign of sine and cosine will change depending on which quadrant we are in. Trigonometry can be fun! Keep practicing, and you'll ace it in no time. If you have any questions or want to go through more examples, just ask! Now go out there and conquer those trig problems!