# Find the Value of tan 135°: A Comprehensive Guide
Hello there! Are you trying to figure out the value of *tan 135°*? Don't worry, we're here to help! In this article, we will break down the problem step-by-step, ensuring you understand not just the answer, but also the underlying concepts. Let's dive in!
## Correct Answer
**The value of tan 135° is -1.**
## Detailed Explanation
To understand why *tan 135° = -1*, we need to break down the concept of the tangent function and how it relates to angles in the unit circle. We will cover the unit circle, reference angles, and the properties of the tangent function in different quadrants.
### Key Concepts
* **Unit Circle:** The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. It's a fundamental tool for understanding trigonometric functions.
* **Tangent Function:** In a unit circle, for any angle θ, the tangent function, *tan θ*, is defined as the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Mathematically, *tan θ = y/x*.
* **Reference Angle:** The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps simplify trigonometric calculations by relating angles in different quadrants to angles in the first quadrant.
* **Quadrants:** The Cartesian plane is divided into four quadrants, numbered I to IV, going counter-clockwise. The signs of the x and y coordinates vary in each quadrant, which affects the signs of trigonometric functions.
### Step-by-Step Breakdown
1. **Locate 135° on the Unit Circle:**
* The angle 135° is in the second quadrant. In the unit circle, start at the positive x-axis (0°) and rotate counter-clockwise. An angle of 135° lies 45° beyond the y-axis (90°).
2. **Find the Reference Angle:**
* To find the reference angle for 135°, subtract it from 180° (since it's in the second quadrant).
* Reference angle = 180° - 135° = 45°
3. **Determine the Coordinates on the Unit Circle:**
* For a 45° angle in the first quadrant, the coordinates on the unit circle are (√2/2, √2/2).
* In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive. Therefore, the coordinates for 135° are (-√2/2, √2/2).
4. **Calculate tan 135°:**
* *tan θ = y/x*
* *tan 135° = (√2/2) / (-√2/2)*
* *tan 135° = -1*
Therefore, *tan 135° = -1*.
### Understanding the Tangent Function in Different Quadrants
To further clarify, let's briefly discuss the behavior of the tangent function in all four quadrants:
* **Quadrant I (0° - 90°):** In the first quadrant, both x and y coordinates are positive. Therefore, *tan θ* is positive.
* **Quadrant II (90° - 180°):** In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive. Therefore, *tan θ* is negative.
* **Quadrant III (180° - 270°):** In the third quadrant, both x and y coordinates are negative. Therefore, *tan θ* is positive.
* **Quadrant IV (270° - 360°):** In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. Therefore, *tan θ* is negative.
This pattern helps us remember that *tan 135°*, being in the second quadrant, must be negative.
### Alternative Method: Using Trigonometric Identities
Another way to find the value of *tan 135°* is by using trigonometric identities. We can express 135° as the sum of two angles for which we know the tangent values. For instance, 135° = 90° + 45°.
However, since *tan 90°* is undefined, we can't directly use the tangent addition formula *tan(A + B) = (tan A + tan B) / (1 - tan A tan B)*. Instead, we can use the identity *tan(180° - θ) = -tan θ*.
1. **Apply the Identity:**
* *tan 135° = tan(180° - 45°)*
* Using the identity, *tan(180° - 45°) = -tan 45°*
2. **Evaluate tan 45°:**
* We know that *tan 45° = 1*
3. **Substitute:**
* *tan 135° = -tan 45° = -1*
Thus, using trigonometric identities, we again find that *tan 135° = -1*.
### Real-World Applications of Tangent
The tangent function is not just a theoretical concept; it has numerous applications in various fields:
* **Navigation:** Used in calculating angles and distances.
* **Physics:** Essential in mechanics for analyzing forces and motion, especially on inclined planes.
* **Engineering:** Critical in designing structures, calculating slopes, and determining angles of elevation.
* **Computer Graphics:** Used in transformations, rotations, and projections in 3D modeling.
Understanding the tangent function and its properties is crucial for anyone studying these fields.
### Common Mistakes to Avoid
When calculating trigonometric functions, especially for angles outside the range of 0° to 90°, it's easy to make mistakes. Here are a few common pitfalls to avoid:
* **Incorrectly Identifying the Quadrant:** Ensure you correctly identify the quadrant in which the angle lies. This affects the sign of the trigonometric function.
* **Using the Wrong Reference Angle:** Always calculate the reference angle correctly. For angles in the second quadrant, subtract the angle from 180°; in the third quadrant, subtract 180° from the angle; and in the fourth quadrant, subtract the angle from 360°.
* **Forgetting the Sign:** Pay close attention to the sign of the trigonometric function based on the quadrant. For example, tangent is negative in the second and fourth quadrants.
* **Confusing Sine, Cosine, and Tangent:** Make sure you know the correct definitions: *sin θ = y/r*, *cos θ = x/r*, and *tan θ = y/x*, where *r* is the radius (in the unit circle, *r = 1*).
By avoiding these common mistakes, you can improve your accuracy and understanding of trigonometric functions.
## Key Takeaways
* The value of *tan 135°* is **-1**.
* *tan 135°* is in the second quadrant, where the tangent function is negative.
* The reference angle for 135° is 45°.
* You can find *tan 135°* using the unit circle, reference angles, or trigonometric identities.
* The tangent function has many real-world applications in fields like navigation, physics, engineering, and computer graphics.
We hope this explanation has clarified how to find the value of *tan 135°*. Keep practicing, and you'll master these concepts in no time!
