Finding Sine And Cosine Values Using Trigonometric Identities

Hey guys! Today, we're diving into a fun trigonometry problem where we'll use trigonometric identities to figure out the values of sine and cosine for an angle θ. The cool part is that we're given the value of cos(2θ) and a range for θ. So, let's put on our math hats and get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what we're dealing with. We're given that cos(2θ) = -12/13, and θ lies between 90° and 180°. This means θ is in the second quadrant. Why is this important? Well, in the second quadrant, sine is positive, and cosine is negative. This knowledge will be crucial later when we determine the signs of our answers. The main goal here is to find the exact values of sin(θ) and cos(θ).

Key Trigonometric Identities

To solve this, we'll need to arm ourselves with some powerful trigonometric identities. Specifically, we'll use the double-angle formulas for cosine. There are a couple of ways to express cos(2θ), but the ones most helpful to us here are:

1. cos(2θ) = 2cos²(θ) - 1
2. cos(2θ) = 1 - 2sin²(θ)

These identities allow us to relate cos(2θ) to cos(θ) and sin(θ) individually. This is exactly what we need to find our target values. These identities are derived from the angle addition formulas, which are fundamental in trigonometry. They provide a bridge between multiple angles and single angles, allowing us to simplify complex expressions and solve equations. Mastering these identities is super important for anyone delving deeper into trigonometry and calculus. Remember, practice makes perfect, so try using them in various problems to get comfortable with their application.

Finding cos(θ)

Let's start by finding cos(θ). We'll use the identity:

cos(2θ) = 2cos²(θ) - 1

We know cos(2θ) = -12/13, so we can plug that in:

-12/13 = 2cos²(θ) - 1

Now, let's solve for cos²(θ). First, add 1 to both sides:

1 - 12/13 = 2cos²(θ)

1/13 = 2cos²(θ)

Next, divide both sides by 2:

1/26 = cos²(θ)

Now, take the square root of both sides:

cos(θ) = ±√(1/26)

cos(θ) = ±1/√26

We can rationalize the denominator by multiplying both the numerator and denominator by √26:

cos(θ) = ±√26 / 26

Remember earlier when we determined that θ is in the second quadrant, where cosine is negative? That means we choose the negative root:

cos(θ) = -√26 / 26

Great! We've found cos(θ). This step involved a bit of algebraic manipulation, but it's all about isolating the term we want. The key here was recognizing which identity to use and then carefully following the steps to solve for cos(θ). Remember to always consider the sign based on the quadrant in which the angle lies. This is a common pitfall, so always double-check! We're halfway there, guys! Now, let's tackle sin(θ).

Finding sin(θ)

Now, let's find sin(θ). We'll use the other form of the double-angle identity:

cos(2θ) = 1 - 2sin²(θ)

Again, plug in cos(2θ) = -12/13:

-12/13 = 1 - 2sin²(θ)

Let's solve for sin²(θ). First, subtract 1 from both sides:

-12/13 - 1 = -2sin²(θ)

-25/13 = -2sin²(θ)

Now, divide both sides by -2:

25/26 = sin²(θ)

Take the square root of both sides:

sin(θ) = ±√(25/26)

sin(θ) = ±5/√26

Rationalize the denominator:

sin(θ) = ±5√26 / 26

Since θ is in the second quadrant, sine is positive, so we choose the positive root:

sin(θ) = 5√26 / 26

Awesome! We've found sin(θ). Just like before, this involved careful algebraic manipulation. Recognizing the correct identity and paying attention to the signs are crucial. The process is very similar to finding cos(θ), but it reinforces the importance of understanding the different forms of the double-angle identities. We're on a roll, guys! We've successfully found both sin(θ) and cos(θ).

Summarizing Our Results

Let's recap what we've found:

• cos(θ) = -√26 / 26
• sin(θ) = 5√26 / 26

These are the exact values of cosine and sine for the angle θ, given the initial conditions. We used the double-angle formula for cosine, some algebra, and our knowledge of quadrants to solve this problem. Remember, the key to success in trigonometry is understanding the identities and knowing how to apply them. We successfully navigated through the problem by breaking it down into smaller steps. First, we identified the relevant identities. Then, we carefully solved for cos(θ) and sin(θ), paying close attention to the signs based on the quadrant. Finally, we summarized our results. This step-by-step approach is a powerful strategy for tackling any math problem.

Practice Makes Perfect

Trigonometry can seem tricky at first, but the more you practice, the easier it becomes. Try working through similar problems, and don't be afraid to make mistakes – that's how we learn! The journey of mastering trigonometry is like building a house; each concept is a brick, and the more bricks you lay, the stronger your foundation becomes. Start with the basics, like understanding the unit circle and the definitions of trigonometric functions. Then, move on to more complex topics like identities and equations. Don't rush the process; take your time to truly understand each concept before moving on. And remember, there are tons of resources available to help you, from textbooks and online tutorials to math forums and study groups.

Conclusion

So, there you have it! We've successfully used trigonometric identities to find the values of sin(θ) and cos(θ) when given cos(2θ) and the range of θ. Remember the key steps: identify the relevant identities, substitute the given values, solve for the desired trigonometric functions, and consider the quadrant to determine the correct sign. Keep practicing, and you'll become a trigonometry pro in no time! Keep up the awesome work, guys, and I'll catch you in the next math adventure!