Tan 135 Degrees Value: Simple Explanation

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Hello! You've asked about the value of tan 135 degrees. Don't worry, I'm here to provide you with a clear, step-by-step explanation so you can understand it easily. We'll break down the concept and find the correct answer together.

Correct Answer

The value of tan 135 degrees is -1.

Detailed Explanation

Now, let's dive into why tan 135 degrees equals -1. We'll cover the unit circle, trigonometric functions, and how to apply these concepts to find the value. It might seem tricky at first, but we'll break it down into simple, manageable parts. Think of trigonometry as a puzzle – once you understand the pieces, it all comes together!

Key Concepts

Before we get started, let's refresh some essential trigonometric concepts:

  • Trigonometric Functions: These functions (sine, cosine, tangent, etc.) relate angles of a triangle to the ratios of its sides. The tangent function (tan) is specifically the ratio of the opposite side to the adjacent side in a right-angled triangle.
  • Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in a coordinate plane. It’s a crucial tool for understanding trigonometric functions because it allows us to visualize angles and their corresponding sine, cosine, and tangent values.
  • Angles in Standard Position: An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. Positive angles are measured counterclockwise, and negative angles are measured clockwise.
  • Reference Angles: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. Reference angles help simplify trigonometric calculations for angles greater than 90 degrees.

The Unit Circle and Tan

The unit circle is our best friend when it comes to understanding trigonometric values. Here’s how it works:

  1. Plotting the Angle: Imagine an angle of 135 degrees in standard position on the unit circle. Start at the positive x-axis (0 degrees) and rotate counterclockwise 135 degrees.
  2. Coordinates: The point where the terminal side of the 135-degree angle intersects the unit circle has coordinates. Let's call this point P. These coordinates are related to the cosine and sine of the angle.
  3. Cosine and Sine: The x-coordinate of point P represents the cosine of 135 degrees (cos 135°), and the y-coordinate represents the sine of 135 degrees (sin 135°).
  4. Tangent: The tangent of an angle is the ratio of sine to cosine: tan θ = sin θ / cos θ. So, tan 135° = sin 135° / cos 135°.

Finding the Coordinates for 135 Degrees

Now, let's determine the coordinates of point P on the unit circle for a 135-degree angle.

  1. Reference Angle: The reference angle for 135 degrees is 180 degrees - 135 degrees = 45 degrees. This means the triangle formed by the terminal side of the 135-degree angle, the x-axis, and a perpendicular line to the x-axis is a 45-45-90 right triangle.
  2. 45-45-90 Triangle: In a 45-45-90 triangle, the sides are in the ratio 1:1:√2. Since we're on the unit circle (radius = 1), the sides opposite the 45-degree angles are 1/√2 (or √2/2 when rationalized). To make calculations easier, let's use √2/2.
  3. Coordinates in Quadrant II: In the second quadrant (where 135 degrees lies), x-coordinates are negative, and y-coordinates are positive. Therefore, the coordinates of point P are (-√2/2, √2/2).

So, cos 135° = -√2/2 and sin 135° = √2/2.

Calculating Tan 135 Degrees

Now that we have the sine and cosine values, we can calculate tan 135°:

tan 135° = sin 135° / cos 135° tan 135° = (√2/2) / (-√2/2) tan 135° = -1

Therefore, the value of tan 135 degrees is indeed -1.

Visualizing with a Graph

It can also be helpful to visualize the tangent function's graph. The tangent function has a period of 180 degrees, meaning its values repeat every 180 degrees. It's positive in the first and third quadrants and negative in the second and fourth quadrants.

  • At 45 degrees (first quadrant), tan 45° = 1.
  • At 135 degrees (second quadrant), tan 135° = -1.
  • At 225 degrees (third quadrant), tan 225° = 1 (135 + 90 = 225, so it repeats with opposite sign)
  • At 315 degrees (fourth quadrant), tan 315° = -1.

The graph confirms that tan 135 degrees is -1.

Step-by-Step Breakdown:

Let's recap the process in clear steps:

  1. Identify the angle: We're working with 135 degrees.
  2. Find the reference angle: 180 degrees - 135 degrees = 45 degrees.
  3. Determine the coordinates on the unit circle: For a 45-degree reference angle in the second quadrant, the coordinates are (-√2/2, √2/2).
  4. Identify sine and cosine values: cos 135° = -√2/2, sin 135° = √2/2.
  5. Calculate tangent: tan 135° = sin 135° / cos 135° = (√2/2) / (-√2/2) = -1.

Key Takeaways

Here’s a quick review of what we’ve learned:

  • The value of tan 135 degrees is -1.
  • We can find this value using the unit circle and understanding the relationship between sine, cosine, and tangent.
  • The reference angle helps simplify trigonometric calculations.
  • Visualizing the tangent function's graph can provide further clarity.

I hope this explanation has helped you understand why tan 135 degrees equals -1. Trigonometry might seem complex, but with a step-by-step approach and a solid grasp of key concepts, it becomes much more manageable. Keep practicing, and you'll master it in no time!