Solving Sin 120° + Cos 150° – Tan 135°: A Geometry Problem

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Hey guys! Today, we're diving into a fun geometry problem that involves trigonometric functions. Specifically, we're going to figure out the result of the operation: sin 120° + cos 150° – tan 135°. Don't worry if it looks intimidating at first; we'll break it down step by step. So, grab your thinking caps, and let's get started!

Understanding the Basics

Before we jump into the main problem, let's quickly recap some essential trigonometric concepts. This will help us tackle the problem more effectively.

  • Sine (sin): In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Think of it as sin(angle) = Opposite / Hypotenuse.
  • Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. Remember it as cos(angle) = Adjacent / Hypotenuse.
  • Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Easy to recall as tan(angle) = Opposite / Adjacent.

These definitions are crucial for understanding how these functions work in different quadrants of the unit circle. Speaking of which...

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It's a fantastic tool for visualizing trigonometric functions for angles beyond the basic 0° to 90° range. Here’s why it’s so useful:

  • Angles: We measure angles counterclockwise from the positive x-axis.
  • Coordinates: For any angle θ, the coordinates of the point where the terminal side of the angle intersects the unit circle are (cos θ, sin θ).
  • Quadrants: The unit circle is divided into four quadrants, each with different sign conventions for sine, cosine, and tangent.

Understanding these sign conventions is super important for solving our problem. Let’s break it down:

  • Quadrant I (0°–90°): All trigonometric functions (sin, cos, tan) are positive.
  • Quadrant II (90°–180°): Sine (sin) is positive, while cosine (cos) and tangent (tan) are negative.
  • Quadrant III (180°–270°): Tangent (tan) is positive, while sine (sin) and cosine (cos) are negative.
  • Quadrant IV (270°–360°): Cosine (cos) is positive, while sine (sin) and tangent (tan) are negative.

With these basics in mind, we're well-equipped to tackle our problem!

Breaking Down the Problem

Now, let's dive into the expression sin 120° + cos 150° – tan 135°. We'll evaluate each term separately and then combine the results.

1. Evaluating sin 120°

First, we need to find the value of sin 120°. Since 120° lies in Quadrant II, where sine is positive, we can use the reference angle to find the value. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. For 120°, the reference angle is 180° - 120° = 60°.

We know that sin 60° = √3 / 2. Since sine is positive in Quadrant II, we have:

sin 120° = sin 60° = √3 / 2

2. Evaluating cos 150°

Next up is cos 150°. Again, 150° is in Quadrant II, but this time, cosine is negative. The reference angle is 180° - 150° = 30°.

We know that cos 30° = √3 / 2. However, since cosine is negative in Quadrant II, we have:

cos 150° = -cos 30° = -√3 / 2

3. Evaluating tan 135°

Finally, let's find tan 135°. The angle 135° is in Quadrant II, where tangent is also negative. The reference angle is 180° - 135° = 45°.

We know that tan 45° = 1. Since tangent is negative in Quadrant II, we have:

tan 135° = -tan 45° = -1

Combining the Results

Now that we've found the values of each term, we can plug them back into the original expression:

sin 120° + cos 150° – tan 135° = (√3 / 2) + (-√3 / 2) – (-1)

Let's simplify this:

(√3 / 2) - (√3 / 2) + 1 = 0 + 1 = 1

So, the result of the operation is 1.

Key Takeaways

Let's recap the main points we covered in solving this problem:

  • Understanding Trigonometric Functions: Knowing the definitions of sine, cosine, and tangent is crucial.
  • The Unit Circle: The unit circle helps visualize trigonometric functions for all angles.
  • Reference Angles: Using reference angles simplifies calculations for angles outside the 0° to 90° range.
  • Quadrant Signs: Remember the sign conventions for each trigonometric function in each quadrant.

By keeping these concepts in mind, you'll be well-prepared to tackle similar trigonometric problems!

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Evaluate cos 210° + sin 330°
  2. Find the value of tan 315° – cos 120°
  3. What is the result of sin 225° + cos 300° – tan 150°?

Try solving these on your own, and feel free to discuss your solutions with friends or classmates. Practice makes perfect, guys!

Conclusion

We've successfully solved the problem sin 120° + cos 150° – tan 135° by breaking it down into smaller, manageable steps. By understanding the basics of trigonometric functions, the unit circle, and reference angles, you can confidently tackle similar problems. Remember, geometry and trigonometry can be super fun once you grasp the fundamentals. Keep practicing, and you'll become a pro in no time!

If you have any questions or want to discuss more geometry problems, feel free to drop a comment below. Happy problem-solving!