How to Find Cos 5pi Over 4 Without A Calculator

Calculating trigonometric functions without a calculator requires understanding of the unit circle, reference angles, and the properties of trigonometric functions. This guide explains how to find cos(5π/4) using these fundamental concepts.

Understanding the Angle 5π/4

The angle 5π/4 radians is equivalent to 405 degrees. To find its position on the unit circle, we can subtract full rotations (2π radians or 360 degrees) until we get an equivalent angle between 0 and 2π.

5π/4 - 2π = 5π/4 - 8π/4 = -3π/4

But -3π/4 is equivalent to 2π - 3π/4 = 5π/4, which brings us back to the original angle. This means 5π/4 is in the third quadrant of the unit circle.

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 angles in the third quadrant, the reference angle is calculated as:

Reference angle = π - (angle - π) = 2π - angle

For 5π/4: Reference angle = 2π - 5π/4 = 8π/4 - 5π/4 = 3π/4

So, the reference angle for 5π/4 is 3π/4 (135 degrees).

Determining the Sign of Cosine

The sign of cosine depends on the quadrant of the angle:

First quadrant (0 to π/2): cos(θ) is positive
Second quadrant (π/2 to π): cos(θ) is negative
Third quadrant (π to 3π/2): cos(θ) is negative
Fourth quadrant (3π/2 to 2π): cos(θ) is positive

Since 5π/4 is in the third quadrant, cos(5π/4) is negative.

Calculating cos(5π/4)

Using the reference angle and the properties of cosine in the third quadrant:

cos(5π/4) = -cos(3π/4)

We know that cos(3π/4) = -√2/2 (from the 45-45-90 triangle in the second quadrant)

Therefore, cos(5π/4) = -(-√2/2) = √2/2

However, this contradicts our earlier determination that cosine should be negative in the third quadrant. Let's correct this:

Actually, cos(5π/4) = -cos(3π/4) = -(-√2/2) = √2/2. This is correct because cosine is negative in the third quadrant, and -(-√2/2) = √2/2 is positive.

The correct calculation is:

cos(5π/4) = -cos(3π/4) = -(-√2/2) = √2/2

Verification with Known Values

We can verify our result by using the cosine of a sum formula:

cos(5π/4) = cos(π + π/4) = -cos(π/4) = -√2/2

This confirms that cos(5π/4) is indeed -√2/2, not √2/2. The earlier mistake was in interpreting the sign rule for cosine in the third quadrant.

The correct value is cos(5π/4) = -√2/2. The reference angle approach was correct, but the final sign determination needed correction.

Frequently Asked Questions

Why is cos(5π/4) negative?

Because 5π/4 (405 degrees) is in the third quadrant where cosine values are negative. The reference angle approach confirms this.

How do I find the reference angle for 5π/4?

The reference angle is calculated as 2π - 5π/4 = 3π/4 (135 degrees).

What is the exact value of cos(5π/4)?

The exact value is -√2/2. This comes from cos(π + π/4) = -cos(π/4).

Can I use a calculator to verify this result?

Yes, entering "cos(5π/4)" in a calculator should return approximately -0.7071, which matches -√2/2.