Calculating Trig Values: Cosine, Cosecant, And Tangent

by Andrew McMorgan 55 views

Hey there, math enthusiasts! 👋 Today, we're diving into the world of trigonometry, specifically focusing on how to find the exact values of cosine, cosecant, and tangent when given a point on the terminal side of an angle. Let's get started, shall we?

Understanding the Basics: Trigonometric Functions

Before we jump into the problem, let's quickly recap the fundamental trigonometric functions. Remember those from your math class? They're super important! We'll be using these concepts a lot.

  • Cosine (cos θ): This relates the adjacent side to the hypotenuse in a right triangle. It's defined as cos θ = adjacent / hypotenuse.
  • Cosecant (csc θ): This is the reciprocal of the sine function. It relates the hypotenuse to the opposite side. It's defined as csc θ = 1 / sin θ = hypotenuse / opposite.
  • Tangent (tan θ): This relates the opposite side to the adjacent side. It's defined as tan θ = opposite / adjacent.

Got it? Great! Now, let's look at what we're given and how to use this information. Our starting point is the point (-2, -7). This point lies on the terminal side of an angle θ. This means that if we were to draw this angle in the standard position (with its vertex at the origin and initial side along the positive x-axis), the terminal side would pass through the point (-2, -7). From this point, we will be able to find the exact values of cos θ, csc θ, and tan θ.

Now, let's break down how to solve this problem step-by-step. Follow along, and you'll become a trigonometry pro in no time! Remember, practice makes perfect, so don't be afraid to try more problems on your own.

Step-by-Step Solution: Finding the Trig Values

Alright, let's get down to business! Here's how we find the exact values for cos θ, csc θ, and tan θ when the point (-2, -7) lies on the terminal side of the angle θ. We'll start with the coordinates of the point, that is crucial here, guys!

  1. Visualize the Point: Imagine the point (-2, -7) on the Cartesian plane. The x-coordinate is -2, and the y-coordinate is -7. This point lies in the third quadrant. This is really important because the signs of our trig functions will depend on the quadrant in which the terminal side of θ lies.

  2. Form a Right Triangle: Draw a right triangle using the point (-2, -7), the x-axis, and the y-axis. The legs of the right triangle will have lengths equal to the absolute values of the coordinates of the point. The horizontal leg (adjacent side) has a length of |-2| = 2, and the vertical leg (opposite side) has a length of |-7| = 7. The hypotenuse, which we'll call r, can be found using the Pythagorean theorem.

  3. Calculate the Hypotenuse (r): Use the Pythagorean theorem to find the length of the hypotenuse: r² = x² + y². In our case, r² = (-2)² + (-7)² = 4 + 49 = 53. So, r = √53. The hypotenuse is always positive.

  4. Find cos θ: Remember, cos θ = adjacent / hypotenuse. The adjacent side is the x-coordinate (-2), and the hypotenuse is √53. So, cos θ = -2 / √53. To rationalize the denominator, we multiply the numerator and denominator by √53, which gives us cos θ = (-2√53) / 53.

  5. Find csc θ: The cosecant is the reciprocal of sine, and sin θ = opposite / hypotenuse. The opposite side is the y-coordinate (-7), and the hypotenuse is √53. So, sin θ = -7 / √53. Therefore, csc θ = 1 / sin θ = √53 / -7, or csc θ = (-√53) / 7.

  6. Find tan θ: tan θ = opposite / adjacent. The opposite side is -7, and the adjacent side is -2. Thus, tan θ = -7 / -2 = 7 / 2. This is a positive value, which makes sense since the tangent is positive in the third quadrant.

Easy peasy, right? Now, you've successfully calculated all the trigonometric functions using the point on the terminal side of an angle. Let's wrap things up with a nice summary!

Summary of the Trigonometric Values

Let's put all the values together so we can easily see the results. Guys, we've successfully found all the requested trigonometric values!

Given that the point (-2, -7) lies on the terminal side of angle θ, we've determined the following:

  • cos θ = (-2√53) / 53
  • csc θ = (-√53) / 7
  • tan θ = 7 / 2

See? Not so hard, after all! Trigonometry might seem a little intimidating at first, but with some practice and a good understanding of the basics, you'll be acing these problems in no time. If you got stuck at any point, go back and review the steps, or try working through a similar problem on your own. You got this!

Tips for Mastering Trigonometry

Want to become a trigonometry superstar? Here are a few extra tips and tricks to help you on your journey:

  • Practice, practice, practice: The more problems you solve, the more comfortable you'll become with the concepts.
  • Understand the unit circle: Familiarize yourself with the unit circle. It's a powerful tool for understanding trigonometric functions and their relationships.
  • Memorize key formulas: Knowing the basic formulas like the Pythagorean theorem and the definitions of sine, cosine, and tangent is essential.
  • Visualize: Draw diagrams! Visualizing the problem can often make it easier to understand and solve.
  • Review: Don't be afraid to revisit the basics. Sometimes, a quick review of the fundamentals can clear up any confusion.

Keep practicing, keep learning, and you'll be well on your way to mastering trigonometry! You got this! 👍