How to Find Csc90 Without A Calculator

What is csc(θ)?

The cosecant function, often written as csc(θ), is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function:

The cosecant function is periodic with a period of 360°, meaning it repeats its values every full rotation. It is undefined when sin(θ) equals zero, which occurs at θ = 0°, 180°, 360°, etc.

What is csc(90°)?

csc(90°) represents the cosecant of 90 degrees. To find its value, we need to evaluate sin(90°) and then take its reciprocal. This is a fundamental value in trigonometry that appears in many mathematical and scientific applications.

How to Calculate csc(90°)

Calculating csc(90°) involves two main steps:

1. Determine the value of sin(90°).
2. Take the reciprocal of sin(90°) to find csc(90°).

From the unit circle, we know that sin(90°) = 1. Therefore, csc(90°) = 1 / sin(90°) = 1 / 1 = 1.

Step-by-Step Calculation

1. Step 1: Recall that sin(90°) = 1. This is a standard trigonometric value derived from the unit circle.
2. Step 2: Apply the cosecant function definition: csc(θ) = 1 / sin(θ).
3. Step 3: Substitute θ = 90°: csc(90°) = 1 / sin(90°) = 1 / 1 = 1.

The result is straightforward because sin(90°) is a well-known value in trigonometry.

Worked Example

Let's verify the calculation with an example:

The calculation confirms that csc(90°) equals 1.