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Sec 90

The secant angle in a right-angle triangle is defined as the ratio of its hypotenuse and adjacent side. For example, consider a right-angled triangle ABC with angle θ made between the hypotenuse and adjacent side; then sec θ is given by:
Sec θ = hypotenuse / adjacent side.

Alternatively, secant is the reciprocal of the cosine function. Therefore,
sec θ = 1/cos θ
For sec 90, the angle 90 lies in the positive y-axis, which gives an undefined value.
Since secant is a periodic function, we can represent sec 90 as:
Sec (90 + 360n), where n belongs to integers.
3. Also, secant is an even function, the value of sec 90 = sec (-90) = undefined.

Value of sec 90, when 90 is in degrees

The value of sec 90 is undefined. It means the value does not exist. The value of sec 90 = 1/0, which is a contradictory value and denotes infinite value (∞). We can find the value of sec 90 in two ways:
Method 1: We can find the value of sec 90 from the cosine function. We know cosine 90 = 0. Therefore,
Sec 90 =1/cos 90 = 1/0 = undefined
Method 2: We can find the value of sec 90 using a unit of circle

To find the value using a unit of a circle, rotate the circle with radius r in an anticlockwise direction with the positive x-axis. The value of sec 90 is equal to the reciprocal of the x-coordinate point of intersection (0,1).
Therefore, we get the value of sec 90 as 1/x = undefined.

Value of sec 90, when 90 is in radians

We can represent 90 degrees in terms of radians as:
90 degrees = 90° × (π/180°) rad = π/2 = 1.5707
Therefore, sec 90 = sec 1.5707 = undefined.

Value of sec 90 in radians

The value of sec 90 in radians = -2.231776

Sec 90 in trigonometry

Sec 90 in trigonometry can be represented as:

  • ± 1/√(1 - sin²(90°))
  • ± √(1 + tan²(90°))
  • ± √(1 + cot²(90°))/cot 90°
  • ± cosec 90°/√(cosec²(90°) - 1)
  • 1/cos 90°

Using trigonometry identities:

  • -sec(180° - 90°) = -sec 90°
  • -sec(180° + 90°) = -sec 270°
  • cosec(90° + 90°) = cosec 180°
  • cosec(90° - 90°) = cosec 0°

Example 1:
Find the value of Sec (270 – x) Sec (90 – x) – tan (270 – x) tan (90 – x).
Solution:

We need to find the value of Sec (270 – x) Sec (90 – x) – tan (270 – x) tan (90 – x)
= (– cosec x cosec x) – (– cot x cot x)
= -cosec 2 x + cot2 x
= – 1

Example 2:
What is the value of sin 30° + 1/sec 90°?
Solution:

We know that, sec x = 1/cos x or cos x = 1/sec x
Thus, sin 30° + 1/sec 90° = sin 30° + cos 90°
= 1/2 + 0
= 1/2

Example 3:
What is the value of 2 sec (90°) / 3 cosec (90°)?
Solution:

From the trigonometric identities, we know, sec (90°) = undefined (∞) and cosec 90° = 1.
Therefore, the value of 2 sec (90°) / 3 cosec (90°) = undefined.

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