Without Using Calculator Find The Value of Sin 105

Finding the value of sin 105 degrees without a calculator requires using trigonometric identities and properties of angles. This guide explains two reliable methods to determine sin 105° using known values of sine and cosine functions.

Method 1: Using Angle Sum Identity

The angle sum identity for sine states that:

sin(A + B) = sin A cos B + cos A sin B

We can express 105° as the sum of 60° and 45°, both of which have known sine and cosine values:

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

Substitute the known values:

sin(60°) = √3/2 ≈ 0.8660
cos(45°) = √2/2 ≈ 0.7071
cos(60°) = 1/2 = 0.5
sin(45°) = √2/2 ≈ 0.7071

Now 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 two terms together:

sin(105°) = 0.6124 + 0.3536 = 0.9659

Therefore, sin(105°) ≈ 0.9659.

Method 2: Using Reference Angle

First, determine the reference angle for 105°:

Reference angle = 180° - 105° = 75°

Since 105° is in the second quadrant, sine is positive in this quadrant. The reference angle is 75°, but we don't have the exact value of sin(75°) memorized. Instead, we can use the angle sum identity again to find sin(75°):

sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)

Substitute the known values:

sin(45°) = √2/2 ≈ 0.7071
cos(30°) = √3/2 ≈ 0.8660
cos(45°) = √2/2 ≈ 0.7071
sin(30°) = 1/2 = 0.5

Calculate each term:

sin(45°)cos(30°) = (√2/2)(√3/2) = √6/4 ≈ 0.6124

cos(45°)sin(30°) = (√2/2)(1/2) = √2/4 ≈ 0.3536

Add the two terms together:

sin(75°) = 0.6124 + 0.3536 = 0.9659

Now, since 105° is in the second quadrant where sine is positive:

sin(105°) = sin(75°) = 0.9659

Therefore, sin(105°) ≈ 0.9659.

Comparison of Methods

Both methods yield the same result, but they approach the problem differently:

Method	Advantages	Disadvantages
Angle Sum Identity	Direct application of known identities	Requires knowledge of multiple angle values
Reference Angle	Uses familiar reference angles	Requires additional calculation for reference angle

Both methods are valid and can be used interchangeably. The choice between them depends on which trigonometric values you have memorized or are more comfortable working with.

Frequently Asked Questions

Why can't I just use a calculator to find sin(105°)?: This guide demonstrates how to find the value without a calculator, which is useful for understanding trigonometric concepts and verifying calculator results.
Is sin(105°) the same as sin(75°)?: No, sin(105°) is equal to sin(75°) because 105° and 75° are supplementary angles (they add up to 180°), and sine is positive in both the first and second quadrants.
Can I use the angle sum identity for any angle combination?: Yes, the angle sum identity can be applied to any two angles that add up to the angle you're trying to find the sine of, as long as you know the sine and cosine values of the component angles.
What is the exact value of sin(105°)?: The exact value is √6/4 + √2/4, which simplifies to (√6 + √2)/4. The decimal approximation is approximately 0.9659.