How to Find Cos 75 Without A Calculator

Calculating the cosine of 75 degrees (cos 75) without a calculator requires using trigonometric identities and known values of common angles. This guide explains two reliable methods to find cos 75 using fundamental trigonometric identities and the values of sine and cosine for 45° and 30°.

Method 1: Using Angle Sum Identity

The angle sum identity for cosine allows us to express cos(75°) in terms of cosines and sines of 45° and 30°:

Formula: cos(A + B) = cosA cosB - sinA sinB

Let A = 45° and B = 30°

cos(45° + 30°) = cos45° cos30° - sin45° sin30°

We know the following values:

cos45° = √2/2 ≈ 0.7071
cos30° = √3/2 ≈ 0.8660
sin45° = √2/2 ≈ 0.7071
sin30° = 1/2 = 0.5

Plugging these values into the formula:

cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2)

= (√6/4) - (√2/4)

= (√6 - √2)/4 ≈ (2.4495 - 1.4142)/4

= 1.0353/4 ≈ 0.2588

So, cos 75° ≈ 0.2588.

Note: The exact value is (√6 - √2)/4, while the approximate decimal value is 0.2588.

Method 2: Using Half-Angle Formula

We can also find cos 75° using the half-angle formula, but this requires knowing cos 150° first. Here's how:

Step 1: Use the double-angle formula for cosine:

cos(2θ) = 2cos²θ - 1

Let 2θ = 150°, so θ = 75°

cos(150°) = 2cos²(75°) - 1

We know cos(150°) = -cos(30°) = -√3/2 ≈ -0.8660

Step 2: Solve for cos(75°):

-√3/2 = 2cos²(75°) - 1

2cos²(75°) = 1 - √3/2

cos²(75°) = (1 - √3/2)/2

cos(75°) = √[(1 - √3/2)/2]

Calculating this gives the same result as Method 1: cos(75°) ≈ 0.2588.

Comparison of Methods

Both methods yield the same result, but Method 1 is generally more straightforward because it uses known values of common angles (45° and 30°). Method 2 requires solving a quadratic equation, which is more complex.

Method	Complexity	Result
Angle Sum Identity	Moderate	cos(75°) = (√6 - √2)/4
Half-Angle Formula	Higher	cos(75°) = √[(1 - √3/2)/2]

Frequently Asked Questions

Why can't I just memorize cos 75°?

While it's possible to memorize specific trigonometric values, understanding how to derive them using identities is more valuable. It helps you solve problems for angles that aren't commonly memorized.

Is there a simpler way to find cos 75°?

The angle sum identity method is the simplest practical approach for calculating cos 75° without a calculator, as it only requires basic arithmetic and known values of common angles.

Can I use a calculator to verify my result?

Yes, you can use a calculator to verify that cos(75°) ≈ 0.2588. This confirms that your manual calculation using trigonometric identities is correct.