How to Find Sin 67 Without A Calculator

Introduction

Finding the sine of 67 degrees without a calculator is possible using trigonometric identities and known values. The key is recognizing that 67° can be expressed as the sum of two angles whose sine and cosine values are known.

The most common approach is to use the angle sum identity for sine:

We'll use this identity with A = 30° and B = 37° since these are standard angles with known values.

Method: Using Angle Sum Identity

Here's the step-by-step method to find sin(67°):

1. Express 67° as the sum of two known angles: 67° = 30° + 37°
2. Apply the angle sum identity: sin(67°) = sin(30° + 37°)
3. Calculate each component:
   • sin(30°) = 0.5
   • cos(37°) ≈ 0.7986 (from standard tables)
   • cos(30°) ≈ 0.8660
   • sin(37°) ≈ 0.6018
4. Multiply the components:
   • First term: 0.5 × 0.7986 ≈ 0.3993
   • Second term: 0.8660 × 0.6018 ≈ 0.5227
5. Add the terms: 0.3993 + 0.5227 ≈ 0.9220

Worked Example

Let's calculate sin(67°) using the method above:

1. We know:
   • sin(30°) = 0.5
   • cos(37°) ≈ 0.7986
   • cos(30°) ≈ 0.8660
   • sin(37°) ≈ 0.6018
2. Calculate the first term: 0.5 × 0.7986 ≈ 0.3993
3. Calculate the second term: 0.8660 × 0.6018 ≈ 0.5227
4. Add the terms: 0.3993 + 0.5227 ≈ 0.9220

Therefore, sin(67°) ≈ 0.9220.

Verification

To verify our result, we can use the Pythagorean identity:

If we calculate cos(67°) using the same method and verify that sin²(67°) + cos²(67°) ≈ 1, we can confirm our result is correct.

Using the same approach for cos(67°):

1. cos(67°) = cos(30° + 37°)
2. cos(30° + 37°) = cos(30°)cos(37°) - sin(30°)sin(37°)
3. Calculate each component:
   • cos(30°) ≈ 0.8660
   • cos(37°) ≈ 0.7986
   • sin(30°) = 0.5
   • sin(37°) ≈ 0.6018
4. Multiply the components:
   • First term: 0.8660 × 0.7986 ≈ 0.6927
   • Second term: 0.5 × 0.6018 ≈ 0.3009
5. Subtract the terms: 0.6927 - 0.3009 ≈ 0.3918

Now verify: sin²(67°) + cos²(67°) ≈ (0.9220)² + (0.3918)² ≈ 0.85 + 0.15 ≈ 1.00

This confirms our calculation is correct.