Tan 7pi 6 Without Calculator

How to calculate tan(7π/6) without a calculator

The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. For any angle θ, tan(θ) = sin(θ)/cos(θ).

To find tan(7π/6), we need to determine the sine and cosine of 7π/6 radians. The angle 7π/6 is equivalent to 210 degrees, which lies in the third quadrant of the unit circle.

In the third quadrant, both sine and cosine values are negative. The reference angle for 7π/6 is calculated as:

We know the sine and cosine of π/6 from standard trigonometric values:

Since 7π/6 is in the third quadrant, we apply the signs for sine and cosine in the third quadrant:

Now we can calculate tan(7π/6):

The simplified form of tan(7π/6) is √3/3.

Step-by-step calculation

1. Identify the angle: 7π/6 radians (210 degrees)
2. Determine the quadrant: 7π/6 is in the third quadrant (π to 3π/2)
3. Find the reference angle: 7π/6 - π = π/6
4. Recall the sine and cosine of π/6:
   • sin(π/6) = 1/2
   • cos(π/6) = √3/2
5. Apply the signs for the third quadrant:
   • sin(7π/6) = -1/2
   • cos(7π/6) = -√3/2
6. Calculate tan(7π/6) = sin(7π/6)/cos(7π/6) = (-1/2)/(-√3/2) = 1/√3
7. Simplify the result: 1/√3 = √3/3

Verification of the result

To ensure our calculation is correct, we can verify it using the tangent addition formula or by checking the unit circle properties.

The tangent of an angle can also be expressed using the addition formula:

Let's use A = 4π/3 and B = -π/6 (since 4π/3 - π/6 = 7π/6):

Now apply the addition formula:

This confirms our earlier result of √3/3.

Common mistakes to avoid

• Forgetting to apply the correct sign for the quadrant: In the third quadrant, both sine and cosine are negative, so their ratio (tangent) is positive.
• Using the wrong reference angle: Always subtract π for angles between π and 3π/2.
• Simplifying the result incorrectly: Remember that 1/√3 can be rationalized to √3/3.
• Confusing radians with degrees: Ensure you're working with radians throughout the calculation.