Trigonometric Values In Terms Of P: Solving For Sin, Tan, Cos

by SLV Team 62 views

Hey guys! Ever stumbled upon a trig problem that looks like it's written in another language? Well, today we're going to break down one of those problems and make it super easy to understand. We're given that cos 25° = 1/p, and our mission is to find sin 25°, tan 25°, cos 65°, and sin 65° all in terms of p. Sounds like a puzzle, right? Let's dive in and solve it together!

Understanding the Problem

Before we jump into calculations, let’s make sure we’re all on the same page. Trigonometry is all about the relationships between angles and sides of triangles. When we see something like cos 25° = 1/p, it’s telling us something specific about a right-angled triangle. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. So, if we think of a right triangle with a 25° angle, the side next to the 25° angle (adjacent) divided by the longest side (hypotenuse) is equal to 1/p. Knowing this is the key to unlocking the rest of the problem.

Why is this important? Because once we know one trigonometric ratio for an angle, we can often find the others using some fundamental trigonometric identities. These identities are like the secret sauce of trigonometry, and we’ll be using them a lot in this solution. Specifically, we'll be leaning on the Pythagorean identity (sin²θ + cos²θ = 1) and the relationships between trigonometric functions of complementary angles (angles that add up to 90°).

So, in essence, we're not just crunching numbers; we're exploring the interconnectedness of angles and side lengths in triangles. This kind of problem helps us see how trigonometry is more than just memorizing formulas – it's about understanding the relationships and using them to solve problems. Let's get started!

a. Finding sin 25°

Okay, let's kick things off by finding sin 25°. We know that cos 25° = 1/p, and we need to find sin 25°. The magic trick here is using the Pythagorean identity, which states that sin²θ + cos²θ = 1. This identity is a cornerstone of trigonometry, and it's going to help us bridge the gap between cosine and sine.

Here’s how we apply it:

  1. Start with the identity: sin²(25°) + cos²(25°) = 1
  2. Substitute the known value: We know cos 25° = 1/p, so we substitute that into the equation: sin²(25°) + (1/p)² = 1
  3. Simplify: This gives us sin²(25°) + 1/p² = 1
  4. Isolate sin²(25°): Subtract 1/p² from both sides to get sin²(25°) = 1 - 1/p²
  5. Find a common denominator: To combine the terms on the right side, we need a common denominator, which is p². So, we rewrite 1 as p²/p², giving us sin²(25°) = p²/p² - 1/p²
  6. Combine fractions: This simplifies to sin²(25°) = (p² - 1) / p²
  7. Take the square root: To find sin 25°, we take the square root of both sides: sin(25°) = ±√((p² - 1) / p²)
  8. Simplify the square root: We can simplify this further by taking the square root of the denominator: sin(25°) = ±√(p² - 1) / p

Now, here's a crucial point: sine values can be positive or negative depending on the quadrant of the angle. Since 25° is in the first quadrant (between 0° and 90°), where sine is positive, we take the positive root.

Therefore, sin 25° = √(p² - 1) / p

See? It might look complicated at first, but breaking it down step by step makes it totally manageable. We used the Pythagorean identity, some basic algebra, and a little bit of thinking about quadrants to get our answer. Next up, we'll tackle tan 25°!

b. Finding tan 25°

Alright, let’s move on to finding tan 25°. We've already figured out that cos 25° = 1/p and sin 25° = √(p² - 1) / p. Now, how do we link these to tangent? This is where another fundamental trigonometric identity comes into play: tan θ = sin θ / cos θ. In other words, the tangent of an angle is simply the sine of the angle divided by the cosine of the angle.

So, to find tan 25°, we just need to divide our expression for sin 25° by our expression for cos 25°:

  1. Start with the definition: tan(25°) = sin(25°) / cos(25°)
  2. Substitute known values: Plug in the values we found earlier: tan(25°) = [√(p² - 1) / p] / [1/p]
  3. Divide fractions: Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we rewrite the division as multiplication: tan(25°) = [√(p² - 1) / p] * [p/1]
  4. Simplify: Notice that the p in the numerator and the p in the denominator cancel each other out: tan(25°) = √(p² - 1)

And that’s it! We’ve found tan 25° in terms of p. This was a bit more straightforward than finding sin 25° because we could directly apply the definition of tangent. It’s a great example of how knowing your basic trig identities can save you a lot of time and effort.

Therefore, tan 25° = √(p² - 1)

Now that we've got sine and tangent sorted, let's move on to the next part of the problem: finding cos 65° and sin 65°. This is where we’ll bring in the concept of complementary angles to make our lives even easier.

c. Finding cos 65°

Okay, so we need to find cos 65°. Now, we already know the values for trigonometric functions of 25°, and 65° might seem totally unrelated at first glance. But here's a cool trick: 25° and 65° are complementary angles. What does that mean? It means they add up to 90° (25° + 65° = 90°).

Complementary angles have a special relationship when it comes to sine and cosine. Specifically, the cosine of an angle is equal to the sine of its complement, and vice versa. In mathematical terms:

  • cos θ = sin (90° - θ)
  • sin θ = cos (90° - θ)

This is a super useful property that allows us to jump between sine and cosine values for complementary angles.

So, how does this help us? We want to find cos 65°, and we know sin 25°. Notice that 65° is the complement of 25° (since 90° - 25° = 65°). Therefore, we can use the identity cos θ = sin (90° - θ).

Here’s the breakdown:

  1. Apply the complementary angle identity: cos(65°) = sin(90° - 65°)
  2. Simplify: cos(65°) = sin(25°)

Wait a minute... we already know sin 25°! We found it in part (a). So, we can simply substitute the value we found earlier.

  1. Substitute the known value: cos(65°) = √(p² - 1) / p

Boom! We've found cos 65° without having to do any complicated calculations. This is the power of understanding complementary angles and their relationships. It's like finding a shortcut in a maze!

Therefore, cos 65° = √(p² - 1) / p

Now, just one more to go! Let's find sin 65°. And guess what? We're going to use the same complementary angle trick to make it super easy.

d. Finding sin 65°

Last but not least, let's find sin 65°. Just like with cos 65°, we’re going to leverage the relationship between complementary angles. We already know that 25° and 65° are complementary, and we know the trigonometric values for 25°.

This time, we'll use the other complementary angle identity: sin θ = cos (90° - θ). This tells us that the sine of an angle is equal to the cosine of its complement.

So, to find sin 65°, we can rewrite it in terms of cos 25°:

  1. Apply the complementary angle identity: sin(65°) = cos(90° - 65°)
  2. Simplify: sin(65°) = cos(25°)

And just like before, we already know cos 25°! It was given to us right at the beginning of the problem: cos 25° = 1/p.

So, we can directly substitute this value:

  1. Substitute the known value: sin(65°) = 1/p

That’s it! We’ve found sin 65° in a couple of quick steps. The key here was recognizing the complementary angle relationship and using the appropriate identity. This not only saves us time but also reinforces the importance of understanding these fundamental concepts.

Therefore, sin 65° = 1/p

Conclusion: Tying It All Together

Woohoo! We did it! We've successfully found sin 25°, tan 25°, cos 65°, and sin 65° all in terms of p. Let's recap our findings:

  • sin 25° = √(p² - 1) / p
  • tan 25° = √(p² - 1)
  • cos 65° = √(p² - 1) / p
  • sin 65° = 1/p

The big takeaway here is that trigonometry isn't just about memorizing formulas; it's about understanding the relationships between angles and sides in triangles. We used the Pythagorean identity, the definition of tangent, and the concept of complementary angles to solve this problem. Each of these tools helped us connect the dots and find the answers efficiently.

Remember, when you're faced with a tricky trig problem, break it down step by step. Identify what you know, what you need to find, and which identities or relationships can help you bridge the gap. And don't be afraid to draw diagrams – visualizing the problem can make a huge difference!

So, next time you see a problem like this, you'll be ready to tackle it like a pro. Keep practicing, keep exploring, and most importantly, have fun with trigonometry! You've got this!