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Tanpi/6 X Cospi/2 Without Calculator

Reviewed by Calculator Editorial Team

Calculating tan(π/6) × cos(π/2) without a calculator requires using fundamental trigonometric identities and values. This guide explains how to compute the result using known values of trigonometric functions at specific angles.

How to calculate tan(π/6) × cos(π/2)

To calculate tan(π/6) × cos(π/2) without a calculator, you'll need to know the exact values of these trigonometric functions at the given angles. Here's how to approach the calculation:

  1. Recall the exact value of tan(π/6)
  2. Recall the exact value of cos(π/2)
  3. Multiply the two values together

π/6 radians is equivalent to 30 degrees, and π/2 radians is equivalent to 90 degrees. These are standard angles in trigonometry with well-known exact values.

Step-by-step calculation

Let's break down the calculation into clear steps:

  1. First, find tan(π/6):
    • tan(π/6) = sin(π/6) / cos(π/6)
    • sin(π/6) = 1/2
    • cos(π/6) = √3/2
    • Therefore, tan(π/6) = (1/2) / (√3/2) = 1/√3 = √3/3 (after rationalizing)
  2. Next, find cos(π/2):
    • cos(π/2) = 0
  3. Finally, multiply the two results:
    • tan(π/6) × cos(π/2) = (√3/3) × 0 = 0
tan(π/6) × cos(π/2) = (√3/3) × 0 = 0

Using trigonometric identities

The calculation relies on these fundamental trigonometric identities:

  • tan(θ) = sin(θ)/cos(θ)
  • sin(π/6) = 1/2
  • cos(π/6) = √3/2
  • cos(π/2) = 0

These identities are essential for solving trigonometric problems without a calculator. The key insight is that cos(π/2) is exactly 0, which makes the entire product equal to 0.

Worked example

Let's work through a concrete example to see how this calculation applies:

Suppose you need to evaluate tan(π/6) × cos(π/2) in a physics problem involving circular motion. Here's how you would approach it:

  1. Identify that π/6 is 30 degrees and π/2 is 90 degrees
  2. Recall the exact values:
    • tan(30°) = √3/3
    • cos(90°) = 0
  3. Multiply them: √3/3 × 0 = 0
  4. Interpret the result: The product is 0, which might indicate no tangential component in this context

This example shows how understanding exact trigonometric values can simplify complex problems in physics and engineering.

Frequently Asked Questions

Why is tan(π/6) × cos(π/2) equal to 0?
Because cos(π/2) is exactly 0, and any number multiplied by 0 is 0. The value of tan(π/6) doesn't affect the result in this case.
Can I use a calculator to verify this result?
Yes, you can use a calculator to verify that tan(π/6) ≈ 0.577 and cos(π/2) = 0, confirming that their product is 0.
Are there other angles where tan(θ) × cos(π/2) would be non-zero?
No, because cos(π/2) is always 0, making the product 0 regardless of the value of tan(θ).
How does this calculation relate to unit circles?
On the unit circle, π/6 corresponds to the point (√3/2, 1/2), and π/2 corresponds to (0, 1). The tangent is the ratio of y/x, and cosine is x.
Can this method be used for other trigonometric calculations?
Yes, understanding exact values of trigonometric functions at standard angles is fundamental for many trigonometric calculations.