How to Find Sin 105 Without Calculator

Calculating sin(105°) without a calculator requires using trigonometric identities and angle decomposition. This guide explains two reliable methods: angle decomposition and the sum formula. Both approaches use known values of sine and cosine for standard angles to derive the result.

Angle Decomposition Method

The angle decomposition method breaks down 105° into a sum of angles whose sine and cosine values are known. We'll use 60° and 45° as our base angles.

Formula: sin(105°) = sin(60° + 45°)

Step-by-Step Calculation

Express 105° as the sum of 60° and 45°: 105° = 60° + 45°
Use the sine addition formula:

sin(A + B) = sinAcosB + cosAsinB
Plug in the known values:
- sin(60°) = √3/2 ≈ 0.8660
- cos(60°) = 1/2 = 0.5
- sin(45°) = √2/2 ≈ 0.7071
- cos(45°) = √2/2 ≈ 0.7071
Calculate each term:
- sin(60°)cos(45°) = (√3/2)(√2/2) = √6/4 ≈ 0.6124
- cos(60°)sin(45°) = (1/2)(√2/2) = √2/4 ≈ 0.3536
Add the terms together: 0.6124 + 0.3536 = 0.9660

Note: The exact value is √6/4 + √2/4 = (√6 + √2)/4 ≈ 0.9659

Sum Formula Method

The sum formula method uses the sine addition formula directly with 105° expressed as 60° + 45°.

Formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°)

Worked Example

Let's calculate sin(105°) using the sum formula with the same known values:

Start with the formula: sin(105°) = sin(60° + 45°)
Apply the sine addition formula:

sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°)
Substitute the known values:
- sin(60°) = √3/2
- cos(45°) = √2/2
- cos(60°) = 1/2
- sin(45°) = √2/2
Calculate each term:
- (√3/2)(√2/2) = √6/4
- (1/2)(√2/2) = √2/4
Add the terms: √6/4 + √2/4 = (√6 + √2)/4 ≈ 0.9659

Tip: Remember that 105° is in the second quadrant where sine is positive, which matches our positive result.

Comparison of Methods

Both methods yield the same result but approach the problem differently. The angle decomposition method breaks down the angle first, while the sum formula applies the identity directly.

Method	Steps	Complexity	Result
Angle Decomposition	Break down angle, apply formula	Moderate	(√6 + √2)/4
Sum Formula	Direct application	Moderate	(√6 + √2)/4

The exact value (√6 + √2)/4 is approximately 0.9659, which matches the decimal result from both methods.

FAQ

Why can't I just use a calculator for sin(105°)?: While calculators are convenient, understanding the underlying trigonometric identities helps in mathematical reasoning and problem-solving.
Are there other angles I can use to find sin(105°)?: Yes, you could also use 90° + 15° or 75° + 30°, but these would require knowing sin(15°) or sin(75°), which are less common.
Is the result the same for all methods?: Yes, as long as you use the correct trigonometric identities and known values, all valid methods should yield the same result.
Can I use this method for other angles?: Absolutely. The angle decomposition and sum formula methods work for any angle that can be expressed as a sum of standard angles.