Secant of 225 Without Calculator

Calculating the secant of 225 degrees without a calculator requires understanding of trigonometric identities and the unit circle. This guide provides step-by-step methods to find the exact value of sec(225°).

How to Calculate Secant of 225 Degrees

The secant function is the reciprocal of the cosine function. To find sec(225°), we need to determine cos(225°) first, then take its reciprocal.

Formula: sec(θ) = 1 / cos(θ)

There are several methods to find cos(225°):

Using trigonometric identities
Reference angle method
Unit circle approach

Using Trigonometric Identities

We can use the cosine of sum identity to find cos(225°):

Identity: cos(A + B) = cosAcosB - sinAsinB

Let's break down 225° into 180° + 45°:

cos(225°) = cos(180° + 45°) = cos180°cos45° - sin180°sin45°

= (-1)(√2/2) - (0)(√2/2) = -√2/2

Now, find sec(225°):

sec(225°) = 1 / cos(225°) = 1 / (-√2/2) = -2/√2 = -√2 (after rationalizing)

The exact value of sec(225°) is -√2.

Reference Angle Method

225° is in the third quadrant where cosine is negative. The reference angle is:

Reference angle = 225° - 180° = 45°

We know that cos(45°) = √2/2. Since cosine is negative in the third quadrant:

cos(225°) = -cos(45°) = -√2/2

Therefore, sec(225°) = 1 / (-√2/2) = -√2.

Unit Circle Approach

On the unit circle, 225° corresponds to the point (-√2/2, -√2/2). The x-coordinate represents cosine:

cos(225°) = -√2/2

Thus, sec(225°) = 1 / (-√2/2) = -√2.

Frequently Asked Questions

What is the exact value of sec(225°)?: The exact value of sec(225°) is -√2, which is approximately -1.4142.
Is sec(225°) positive or negative?: Sec(225°) is negative because 225° is in the third quadrant where cosine is negative, and secant is the reciprocal of cosine.
How do I rationalize the denominator of sec(225°)?: To rationalize -2/√2, multiply numerator and denominator by √2: (-2/√2) × (√2/√2) = -2√2/2 = -√2.
What is the relationship between secant and cosine?: The secant function is the reciprocal of the cosine function: sec(θ) = 1/cos(θ).
Can I use a calculator to verify sec(225°)?: Yes, entering "sec(225)" in a calculator should give you approximately -1.4142, which matches our exact value of -√2.