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Sin 80 Without Calculator

Reviewed by Calculator Editorial Team

Calculating sin 80° without a calculator requires understanding of trigonometric identities and angles. This guide explains multiple methods to find the sine of 80 degrees accurately, including using known values and angle relationships.

How to calculate sin 80° without a calculator

There are several approaches to find sin 80° without a calculator. The most common methods involve using known trigonometric values and identities. Here's an overview of the techniques:

Note: All angles in this guide are in degrees unless specified otherwise.

Key trigonometric values

First, recall these fundamental sine values:

  • sin 0° = 0
  • sin 30° = 0.5
  • sin 45° = √2/2 ≈ 0.7071
  • sin 60° = √3/2 ≈ 0.8660
  • sin 90° = 1

Using angle relationships

You can use the fact that sin(θ) = cos(90° - θ) to find sin 80°:

sin 80° = cos(90° - 80°) = cos 10°

Now you need to find cos 10°. This can be done using a series expansion or other identities.

Using trigonometric identities

The most precise method without a calculator is using the Taylor series expansion for sine:

sin x = x - x³/3! + x⁵/5! - x⁷/7! + ...

For 80° (which is 80π/180 radians = 4π/9 radians), the series becomes:

sin(4π/9) ≈ 4π/9 - (4π/9)³/6 + (4π/9)⁵/120 - (4π/9)⁷/5040 + ...

Calculating this manually requires knowing π ≈ 3.1416 and performing multiple multiplications and divisions.

Using the sine addition formula

You can also use the sine addition formula:

sin(a + b) = sin a cos b + cos a sin b

For example, sin 80° = sin(60° + 20°). You would need to know or calculate sin 20° and cos 20°.

Step-by-step method

  1. Convert 80° to radians: 80 × π/180 ≈ 1.3963 radians
  2. Use the Taylor series expansion for sine:
    • First term: 1.3963
    • Second term: - (1.3963)³ / 6 ≈ -0.3333
    • Third term: + (1.3963)⁵ / 120 ≈ +0.0149
    • Fourth term: - (1.3963)⁷ / 5040 ≈ -0.0003
  3. Sum the terms: 1.3963 - 0.3333 + 0.0149 - 0.0003 ≈ 1.0776
  4. This is an approximation of sin 80° ≈ 0.9848 (actual value)

For better accuracy, include more terms in the series expansion.

Example calculation

Let's calculate sin 80° using the angle relationship method:

  1. We know sin 80° = cos 10°
  2. Use the Taylor series for cosine:
    • First term: 1
    • Second term: - (10°)² / 2! ≈ -0.0175
    • Third term: + (10°)⁴ / 4! ≈ +0.0003
    • Fourth term: - (10°)⁶ / 6! ≈ -0.000002
  3. Sum the terms: 1 - 0.0175 + 0.0003 - 0.000002 ≈ 0.9828
  4. This approximates cos 10° ≈ 0.9848 (actual value)

Therefore, sin 80° ≈ 0.9848

Visual guide

The following chart shows the sine values for common angles:

This visualization helps understand how sin 80° relates to other common angles.

FAQ

How accurate are these methods?
The Taylor series method provides reasonable accuracy with enough terms. For most practical purposes, 3-4 terms give good results. The angle relationship method is also accurate when using known values.
Can I use a calculator for part of the calculation?
Yes, you can use a calculator to verify intermediate steps or final results. The goal is to understand the process rather than rely entirely on a calculator.
What's the difference between sin and cos?
Sine and cosine are related through the identity sin(θ) = cos(90° - θ). They represent the y and x coordinates of a point on the unit circle, respectively.
Are there other identities I can use?
Yes, you can use half-angle formulas, product-to-sum identities, or sum-to-product identities depending on the angle you're working with.