How to Find Csc 120 Without A Calculator
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Calculating CSC 120 (cosecant of 120 degrees) without a calculator requires understanding trigonometric identities and applying them step-by-step. This guide explains the process clearly with examples and a built-in calculator.
Understanding CSC 120
The cosecant function, written as csc(θ), is the reciprocal of the sine function: csc(θ) = 1/sin(θ). For θ = 120°, we need to find sin(120°) first, then take its reciprocal.
Note: All angles in this guide are in degrees unless specified otherwise.
Using Trigonometric Identities
We can use the angle addition formula for sine to find sin(120°):
This shows that sin(120°) equals sin(60°), which is √3/2.
Step-by-Step Method
- Identify that 120° is in the second quadrant where sine is positive.
- Use the identity sin(180° - θ) = sin(θ) to find sin(120°).
- Calculate sin(60°) as √3/2.
- Take the reciprocal to find csc(120°) = 1/(√3/2) = 2/√3.
- Rationalize the denominator: 2/√3 = (2√3)/3.
Common Angle Values
Here are some common angle values that can help with calculations:
| Angle | Sine | Cosecant |
|---|---|---|
| 0° | 0 | Undefined |
| 30° | 1/2 | 2 |
| 45° | √2/2 | √2 |
| 60° | √3/2 | 2/√3 or 2√3/3 |
| 90° | 1 | 1 |
Example Calculations
Example 1: Calculating csc(120°)
Using the step-by-step method:
- sin(120°) = sin(180° - 60°) = sin(60°) = √3/2
- csc(120°) = 1/(√3/2) = 2/√3
- Rationalized: 2√3/3 ≈ 1.1547
Example 2: Calculating csc(240°)
Following the same process:
- sin(240°) = sin(180° + 60°) = -sin(60°) = -√3/2
- csc(240°) = 1/(-√3/2) = -2/√3
- Rationalized: -2√3/3 ≈ -1.1547
Frequently Asked Questions
Why is csc(120°) positive?
Cosecant is positive in the second quadrant (90° to 180°) because sine is positive in this range. The reciprocal of a positive number is also positive.
How do I rationalize the denominator?
Multiply the numerator and denominator by √3 to eliminate the square root in the denominator: 2/√3 = (2√3)/3.
What's the difference between csc and sec?
Cosecant (csc) is the reciprocal of sine, while secant (sec) is the reciprocal of cosine. They have different signs in different quadrants.
Can I use this method for any angle?
Yes, this method works for any angle by using appropriate trigonometric identities to reduce it to a known angle.