Tan(β) + Sec(β): Step-by-Step Trigonometry Solution
Hey guys! Today, we're diving deep into a super interesting trigonometry problem. We're given a point (20, -21) that lies on the terminal side of an angle β in standard position, and our mission is to find the value of Tan(β) + Sec(β). Sounds like a fun challenge, right? Let's break it down step by step and make sure we understand every little detail along the way. Trigonometry can seem daunting at first, but with a clear approach and a bit of practice, it becomes much easier to handle. So, buckle up, and let’s get started!
Understanding the Basics: Standard Position and Trigonometric Ratios
Before we jump into the calculations, let's quickly recap some fundamental concepts. When we say an angle is in standard position, it means that the vertex of the angle is at the origin (0, 0) of the coordinate plane, and the initial side of the angle lies along the positive x-axis. The terminal side is the side that rotates, and it's where our point (20, -21) comes into play. This point helps us define the angle β and its trigonometric ratios.
Now, let's talk about those trigonometric ratios. Remember SOH CAH TOA? It's a handy mnemonic to remember the basic ratios: Sine (Sin), Cosine (Cos), and Tangent (Tan). But we also have their reciprocals: Cosecant (Csc), Secant (Sec), and Cotangent (Cot). In our case, we need to focus on Tangent and Secant since those are the ones we’re trying to find.
- Tangent (Tan β): This is defined as the ratio of the opposite side (y-coordinate) to the adjacent side (x-coordinate). So, Tan(β) = y/x.
- Secant (Sec β): This is the reciprocal of the cosine function. Cosine is the ratio of the adjacent side (x-coordinate) to the hypotenuse (r), so Secant is the hypotenuse (r) divided by the adjacent side (x-coordinate). Thus, Sec(β) = r/x.
But what’s this hypotenuse (r) we’re talking about? Well, think of our point (20, -21) as forming a right-angled triangle with the x-axis. The hypotenuse is the distance from the origin to our point, which we can calculate using the Pythagorean theorem. This sets the stage for us to find the exact values of Tan(β) and Sec(β) and eventually solve our problem.
Step-by-Step Solution: Calculating Tan(β) and Sec(β)
Okay, guys, let's get our hands dirty with the math! We have the point (20, -21), and we need to find Tan(β) + Sec(β). Here’s how we'll do it:
1. Find the Hypotenuse (r)
As we discussed, we'll use the Pythagorean theorem to find the hypotenuse r. The formula is: r = √(x² + y²). In our case, x = 20 and y = -21.
r = √(20² + (-21)²) = √(400 + 441) = √841 = 29
So, our hypotenuse r is 29. Keep this number handy; we’ll need it for the Secant calculation.
2. Calculate Tan(β)
Remember, Tan(β) = y/x. We have x = 20 and y = -21, so:
Tan(β) = -21/20
That was pretty straightforward, right? Now we have the value of Tan(β).
3. Calculate Sec(β)
Sec(β) = r/x. We found r = 29 and we know x = 20, so:
Sec(β) = 29/20
Great! We've got Sec(β) as well. We're almost there, guys!
4. Find Tan(β) + Sec(β)
Now, the final step: we just need to add Tan(β) and Sec(β):
Tan(β) + Sec(β) = (-21/20) + (29/20)
Since they have the same denominator, we can easily add the numerators:
Tan(β) + Sec(β) = (-21 + 29) / 20 = 8/20
5. Simplify the Result
We can simplify 8/20 by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
8/20 = (8 ÷ 4) / (20 ÷ 4) = 2/5
And there you have it! Tan(β) + Sec(β) = 2/5.
Why This Matters: Real-World Applications of Trigonometry
Okay, so we've solved this problem, which is awesome! But you might be thinking,
