Sec 135 Degrees Without Calculator

What is sec(135°)?

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). For 135 degrees, we need to find the cosine of 135° and then take its reciprocal.

135° is located in the second quadrant of the unit circle, where cosine values are negative. This means sec(135°) will be negative.

How to calculate sec(135°)

To calculate sec(135°) without a calculator, follow these steps:

1. Identify the reference angle for 135°
2. Find the cosine of the reference angle
3. Apply the sign based on the quadrant
4. Take the reciprocal to get sec(135°)

Step-by-step calculation

Step 1: Find the reference angle

The reference angle for 135° is calculated as:

So, the reference angle is 45°.

Step 2: Find cos(45°)

From trigonometric identities, we know:

Step 3: Apply the sign based on the quadrant

Since 135° is in the second quadrant where cosine is negative:

Step 4: Calculate sec(135°)

Now take the reciprocal of cos(135°):

The exact value is -√2, which is approximately -1.4142.

Using reference angle

Another approach is to use the reference angle directly:

1. Recognize that 135° = 180° - 45°
2. Use the identity: cos(180° - θ) = -cos(θ)
3. Apply to secant: sec(135°) = 1/cos(135°) = 1/(-cos(45°)) = -sec(45°)
4. Since sec(45°) = √2, then sec(135°) = -√2

Verification

To verify our result, let's check with a calculator:

• cos(135°) ≈ -0.7071
• sec(135°) ≈ 1/-0.7071 ≈ -1.4142
• -√2 ≈ -1.4142

The values match, confirming our manual calculation is correct.