Tan 5pi 4 Without Calculator

Calculating tan(5π/4) without a calculator requires understanding of trigonometric identities and the unit circle. This guide will walk you through the process step-by-step, showing you how to determine the tangent of 5π/4 radians using fundamental trigonometric principles.

Understanding tan(5π/4)

The tangent function, tan(θ), is defined as the ratio of sine to cosine for a given angle θ. For tan(5π/4), we're looking at the tangent of an angle that's 5π/4 radians, which is equivalent to 405 degrees.

Remember that trigonometric functions are periodic with a period of π (180°), so we can reduce 5π/4 to an equivalent angle between 0 and 2π by subtracting 2π (360°):

5π/4 - 2π = 5π/4 - 8π/4 = -3π/4

However, negative angles can be converted to positive by adding 2π:

-3π/4 + 2π = 5π/4

This shows that 5π/4 is coterminal with -3π/4, but we'll use the positive angle for our calculations.

Finding the Reference Angle

The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. For 5π/4 radians:

Determine the quadrant: 5π/4 is in the third quadrant (π to 3π/2).
Calculate the reference angle: π - (5π/4 - π) = π - π/4 = 3π/4.

The reference angle is 3π/4 (135°), which is in the second quadrant.

Using the Unit Circle

The unit circle approach involves plotting the angle on the unit circle and determining the coordinates of the corresponding point.

Start at the point (1, 0) on the unit circle.
Move counterclockwise 5π/4 radians (405°).
The terminal side of the angle will be in the third quadrant.
The coordinates of the point will be (-√2/2, -√2/2).

Since tan(θ) = y/x, we have:

tan(5π/4) = (-√2/2) / (-√2/2) = 1

This shows that tan(5π/4) = 1.

Using Trigonometric Identities

We can also use trigonometric identities to find tan(5π/4).

The tangent function has a period of π, so:

tan(5π/4) = tan(5π/4 - 2π) = tan(-3π/4)

Since tangent is an odd function:

tan(-3π/4) = -tan(3π/4)

We know that tan(3π/4) = tan(π - π/4) = -tan(π/4) = -1, so:

tan(-3π/4) = -(-1) = 1

Therefore, tan(5π/4) = 1.

Worked Example

Let's verify our result with a concrete example. Suppose we have a right triangle where the angle θ = 5π/4 radians.

First, find the reference angle: 3π/4 (135°).
In a right triangle with angle 135°, the adjacent side is -√2/2 and the opposite side is √2/2.
Therefore, tan(θ) = opposite/adjacent = (√2/2) / (-√2/2) = -1.
But since 5π/4 is in the third quadrant where tangent is positive, we take the absolute value: 1.

This confirms our earlier result that tan(5π/4) = 1.

Frequently Asked Questions

Why is tan(5π/4) equal to 1?

tan(5π/4) equals 1 because the angle 5π/4 radians (405°) is coterminal with -3π/4 radians (-225°). The tangent function is periodic with period π, and the reference angle for 5π/4 is 3π/4 (135°). In the third quadrant, both sine and cosine are negative, so their ratio (tangent) is positive. The reference angle's tangent is -1, but in the third quadrant, tangent becomes positive.

Can I use a calculator to verify this result?

Yes, you can use a calculator to verify that tan(5π/4) equals 1. However, this guide shows you how to calculate it without a calculator using fundamental trigonometric principles.

What is the difference between tan(5π/4) and tan(π/4)?

tan(π/4) equals 1, but tan(5π/4) also equals 1. The key difference is their positions on the unit circle. π/4 is in the first quadrant where both sine and cosine are positive, while 5π/4 is in the third quadrant where both are negative. However, the ratio of sine to cosine remains the same in magnitude but changes sign based on the quadrant.