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:

• cosec 20 degrees
• cosec 90 degrees
• cosec 75 degrees
• cosec 210 degrees
• cosec 60 degrees
• cosec 30 degrees

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.