Solve Tan 25pi 12 Without A Calculator

Calculating tan(25π/12) without a calculator requires understanding of trigonometric identities and angle reduction formulas. This guide provides a step-by-step solution to find the exact value of this tangent function.

Understanding the Problem

The expression tan(25π/12) involves finding the tangent of an angle that's 25π/12 radians. Since π radians equals 180 degrees, we can convert this to degrees for better intuition:

25π/12 radians = (25π/12) × (180°/π) = 337.5°

337.5° is an angle in the fourth quadrant of the unit circle. In the fourth quadrant, tangent values are negative because sine is negative and cosine is positive.

Step-by-Step Solution

Step 1: Reduce the Angle

First, we reduce the angle to an equivalent angle between 0 and 2π radians (0° to 360°).

25π/12 - 2π = 25π/12 - 24π/12 = π/12

So, tan(25π/12) = tan(π/12).

Step 2: Use Tangent of Difference Formula

We can express π/12 as π/3 - π/4 and use the tangent of difference formula:

tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Let A = π/3 and B = π/4.

Step 3: Calculate Individual Tangents

We know:

tan(π/3) = √3

tan(π/4) = 1

Step 4: Apply the Formula

Plugging into the formula:

tan(π/12) = (√3 - 1) / (1 + √3 × 1) = (√3 - 1) / (1 + √3)

Step 5: Rationalize the Denominator

Multiply numerator and denominator by the conjugate of the denominator:

tan(π/12) = [(√3 - 1)(1 - √3)] / [(1 + √3)(1 - √3)]

= [√3 - 3 - 1 + √3] / [1 - (√3)²]

= [2√3 - 4] / (1 - 3)

= [2√3 - 4] / (-2)

= (4 - 2√3)/2

= 2 - √3

Final Result

Therefore, tan(25π/12) = tan(π/12) = 2 - √3.

Verification

To verify our result, let's calculate tan(π/12) using a calculator and compare:

tan(π/12) ≈ tan(15°) ≈ 0.2679

2 - √3 ≈ 2 - 1.732 ≈ 0.268

The values match, confirming our solution is correct.

Common Mistakes

Forgetting to reduce the angle to within 0 to 2π radians
Incorrectly applying the tangent of difference formula
Not rationalizing the denominator properly
Sign errors when working with angles in different quadrants

Frequently Asked Questions

Why is tan(25π/12) equal to tan(π/12)?

The tangent function has a period of π radians, meaning tan(θ) = tan(θ + kπ) for any integer k. Since 25π/12 - 2π = π/12, the two angles have the same tangent value.

How do I know when to use the tangent of difference formula?

The tangent of difference formula is useful when you need to find the tangent of a complex angle that can be expressed as the difference of two simpler angles whose tangents you know.

What's the significance of rationalizing the denominator?

Rationalizing the denominator simplifies the expression and removes any radicals from the denominator, making the result cleaner and easier to work with in further calculations.

Can I use this method for other angles?

Yes, this method can be adapted for other angles by breaking them down into sums or differences of standard angles whose trigonometric values you know.