Evaluate the cosecant function with a calculator
Evaluate the cosecant function with a calculator

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Cosec 45 Degrees

The value of cosec 45 degrees is 1.4142135. . .. Cosec 45 degrees in radians is written as cosec (45° × π/180°), i.e., cosec (π/4) or cosec (0.785398. . .). In this article, we will discuss the methods to find the value of cosec 45 degrees with examples.

• Cosec 45°: √2
• Cosec 45° in decimal: 1.4142135. . .
• Cosec (-45 degrees): -1.4142135. . . or -√2
• Cosec 45° in radians: cosec (π/4) or cosec (0.7853981 . . .)

## What is the Value of Cosec 45 Degrees?

The value of cosec 45 degrees in decimal is 1.414213562. . .. Cosec 45 degrees can also be expressed using the equivalent of the given angle (45 degrees) in radians (0.78539 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 45 degrees = 45° × (π/180°) rad = π/4 or 0.7853 . . .
∴ cosec 45° = cosec(0.7853) = √2 or 1.4142135. . . Explanation:

For cosec 45 degrees, the angle 45° lies between 0° and 90° (First Quadrant). Since cosecant function is positive in the first quadrant, thus cosec 45° value = √2 or 1.4142135. . .
Since the cosecant function is a periodic function, we can represent cosec 45° as, cosec 45 degrees = cosec(45° + n × 360°), n ∈ Z.
⇒ cosec 45° = cosec 405° = cosec 765°, and so on.
Note: Since, cosecant is an odd function, the value of cosec(-45°) = -cosec(45°).

## Methods to Find Value of Cosec 45 Degrees

The cosecant function is positive in the 1st quadrant. The value of cosec 45° is given as 1.41421. . .. We can find the value of cosec 45 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

## Cosec 45 Degrees Using Unit Circle To find the value of cosec 45 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form 45° angle with the positive x-axis.
• The cosec of 45 degrees equals the reciprocal of the y-coordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.

Hence the value of cosec 45° = 1/y = 1.4142 (approx)

## Cosec 45° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cosec 45 degrees as:

• ± 1/√(1-cos²(45°))
• ± √(1 + tan²(45°))/tan 45°
• ± √(1 + cot²(45°))
• ± sec 45°/√(sec²(45°) – 1)
• 1/sin 45°

Note: Since 45° lies in the 1st Quadrant, the final value of cosec 45° will be positive.

We can use trigonometric identities to represent cosec 45° as,

• cosec(180° – 45°) = cosec 135°
• -cosec(180° + 45°) = -cosec 225°
• sec(90° – 45°) = sec 45°
• -sec(90° + 45°) = -sec 135°

☛ Also Check:

## Examples Using Cosec 45 Degrees

1. Example 1: Find the value of 6 cosec(45°)/8 sec(45°).

Solution:

Using trigonometric identities, we know, cosec(45°) = sec(90° – 45°) = sec 45°.

⇒ cosec(45°) = sec(45°)
⇒ Value of 6 cosec(45°)/8 sec(45°) = 3/4

2. Example 2: Using the value of csc 45°, solve: (1 + cot²(45°)).

Solution:

We know, (1 + cot²(45°)) = (csc²(45°)) = 2
⇒ (1 + cot²(45°)) = 2

3. Example 3: Find the value of csc 45° if sin 45° is 0.7071.

Solution:

Since, csc 45° = 1/sin 45°
⇒ csc 45° = 1/0.7071 = 1.4142

## FAQs on Cosec 45 Degrees

### What is Cosec 45 Degrees?

Cosec 45 degrees is the value of cosecant trigonometric function for an angle equal to 45 degrees. The value of cosec 45° is √2 or 1.4142 (approx).

### What is the Exact Value of Cosec 45 Degrees?

The exact value of cosec 45 degrees can be given accurately up to 8 decimal places as 1.41421356 or as √2.

### How to Find the Value of Cosec 45 Degrees?

The value of cosec 45 degrees can be calculated by constructing an angle of 45° with the x-axis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of cosec 45° is equal to the reciprocal of the y-coordinate (0.7071). ∴ cosec 45° = 1.4142.

How to Find Cosec 45° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cosec 45° can be given in terms of other trigonometric functions as:

• ± 1/√(1-cos²(45°))
• ± √(1 + tan²(45°))/tan 45°
• ± √(1 + cot²(45°))
• ± sec 45°/√(sec²(45°) – 1)
• 1/sin 45°

☛ Also check: trigonometry table

What is the Value of Cosec 45 Degrees in Terms of Tan 45°?

We know, using trig identities, we can write cosec 45° as √(1 + tan²(45°))/tan 45°. Here, the value of tan 45° is equal to 1.

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