Tan 135 Without Calculator
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Reviewed by Calculator Editorial Team
Calculating tan(135°) without a calculator requires understanding trigonometric identities and reference angles. This guide explains the formula, step-by-step solution, and practical applications of this calculation.
How to calculate tan(135°)
The tangent of 135 degrees can be found using trigonometric identities. Since 135° is in the second quadrant, we can use the following identity:
tan(180° - θ) = -tan(θ)
This identity shows that the tangent of an angle in the second quadrant is the negative of the tangent of its reference angle. For 135°, the reference angle is 45° (180° - 135° = 45°).
Remember that tan(45°) = 1, so tan(135°) = -tan(45°) = -1.
Step-by-step solution
- Identify the quadrant of 135°: It's in the second quadrant (90° to 180°).
- Find the reference angle: 180° - 135° = 45°.
- Recall that tan(45°) = 1.
- Apply the identity for the second quadrant: tan(135°) = -tan(45°) = -1.
tan(135°) = -tan(45°) = -1
Practical applications
Knowing tan(135°) is useful in various fields:
- Engineering: Calculating angles in structural analysis
- Physics: Determining components of forces in two dimensions
- Computer graphics: Rotating objects in 2D space
- Navigation: Solving triangles in surveying
For example, if you're working with a force vector at 135° to the x-axis, you can determine its vertical component using the tangent function.
Common mistakes
- Forgetting to account for the negative sign in the second quadrant
- Confusing reference angles with the original angle
- Assuming tan(135°) equals tan(45°) without applying the identity
Always verify the quadrant of the angle before applying trigonometric identities.
FAQ
- Why is tan(135°) negative?
- Because 135° is in the second quadrant where tangent values are negative.
- Can I use this method for other angles?
- Yes, this identity works for any angle in the second quadrant (90° to 180°).
- What's the reference angle for 135°?
- The reference angle is 45° (180° - 135°).
- How does this relate to the unit circle?
- The unit circle shows that tan(135°) corresponds to the point (-√2/2, √2/2), where the ratio of y/x gives -1.
- Is tan(135°) the same as tan(-45°)?
- Yes, because tangent is an odd function (tan(-θ) = -tan(θ)).