Without Using A Calculator Find Sin75

Finding sin(75°) without a calculator requires using trigonometric identities and known values of common angles. This guide explains the method, provides a step-by-step calculation, and includes an interactive calculator for verification.

How to Find sin(75°) Without a Calculator

The sine of 75 degrees can be found using the angle addition formula for sine. This method breaks down 75° into the sum of two angles whose sine values are known: 45° and 30°.

Key Formula

sin(A + B) = sinA cosB + cosA sinB

By applying this formula to 75° = 45° + 30°, we can calculate sin(75°) using the known values of sin(45°), cos(45°), sin(30°), and cos(30°).

Angle Addition Formula

The angle addition formula for sine states that the sine of the sum of two angles is equal to the sum of the products of the sine of one angle and the cosine of the other angle.

Formula

sin(A + B) = sinA cosB + cosA sinB

This formula is fundamental in trigonometry and allows us to find the sine of any angle that can be expressed as the sum of two known angles.

Step-by-Step Calculation

Express 75° as the sum of 45° and 30°: 75° = 45° + 30°
Apply the angle addition formula: 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 product:
- 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 products: sin(75°) ≈ 0.6124 + 0.3536 = 0.9659

Note

The exact value of sin(75°) is (√6 + √2)/4 ≈ 0.9659. The approximation using decimal values is very close to the exact value.

Example Calculation

Let's work through an example to find sin(75°):

Example

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

= (√2/2)(√3/2) + (√2/2)(1/2)

= √6/4 + √2/4

= (√6 + √2)/4 ≈ 0.9659

This calculation shows how to derive sin(75°) using the angle addition formula and known trigonometric values.

Frequently Asked Questions

Why can't I just use a calculator to find sin(75°)?: While calculators are convenient, understanding how to find trigonometric values without one helps deepen your knowledge of trigonometric identities and their applications.
What are the known values used in this calculation?: The calculation uses the known values of sin(45°), cos(45°), sin(30°), and cos(30°), which are fundamental in trigonometry.
Is the angle addition formula the only way to find sin(75°)?: No, there are other methods such as using the sine of supplementary angles or half-angle formulas, but the angle addition formula is straightforward for this case.
What is the exact value of sin(75°)?: The exact value is (√6 + √2)/4, which is approximately 0.9659.
Can I use this method for other angles?: Yes, the angle addition formula can be applied to find the sine of any angle that can be expressed as the sum of two known angles.