sec 90° in Trigonometry

Imagine you are standing next to a tall tree, looking straight up at its top. Your line of sight is perfectly vertical, forming a right angle with the ground beneath you. But have you ever wondered what this means in the world of trigonometry? How does the secant function behave when the angle reaches this critical point? In this article, we’ll dive into the concept of sec 90°, unravel its significance, and explore its real-world applications.

Sec 90°Definition & Explanation:

In trigonometry, sec θ (secant) is one of the six trigonometric ratios. It is the reciprocal of cosine, defined as the ratio of the hypotenuse to the adjacent side in a right triangle.
Consider a right-angled triangle ABC, where B is the right angle and θ is an acute angle at C, as shown in the figure.

secθ = Hypotenuse / Adjacent Side

Also Check: Tan 0

Thus, we can express secant in terms of cosine:
secθ = 1 / cosθ

For θ = 90°, substituting into the equation:
sec90° = 1 / cos90°

Since cos 90° = 0, we get:
sec90° = 1 / 0

Mathematically, division by zero is undefined, meaning that sec 90° does not exist.

So, we can conclude:
❖ sec 90° is undefined.
❖ It does not exist.
❖ 1 / 0 is not a valid mathematical expression.

Also Check: Cos 45

What is sec (90° - θ)?

sec (90° – θ) = sec (1 × 90° – θ)
Here, 90° is multiplied by 1, an odd number so sec will change to cosec. Also, 90° – θ comes in the first quadrant. That means, all trigonometry ratios are positive.
So, sec (90° – θ) = cosec θ
Similarly, for sec (90° + θ).
sec (90° + θ) = –cosec θ (In second quadrant sec is negative)

Examples & Applications:

Example 1: Is sec (–90°) = sec (90°)?
Solution: Yes, since secant is an even function, the value of sec (–90°) = sec (90°) = undefined.

Example 2: Solve: 1 + tan²(90°)
Solution: Using the identity:
1 + tan²θ = sec²θ
For θ = 90°, we get:
1 + tan²(90°) = sec²(90°) = undefined

Example 3: What is the value of sec 450°?
Solution: Since the secant function is a periodic function, we can represent sec 90° as,
sec 90° = sec (90° + n × 360°), n ∈ Z.
Thus:
sec 90° = sec 450° = sec 810°, and so on.
Since sec 90° is undefined, therefore, sec 450° is also undefined.

Also Check: Tan 90

Application in Real Life:

• Engineering: Engineers use trigonometry to design strong and stable structures. For example, sec 90° is undefined, which reminds them to be extra careful with calculations to avoid errors.
• Physics: Trigonometry helps explain how waves move and behave. At certain points, like sec 90°, the values become undefined, showing sudden shifts that scientists need to understand.
• Camera Angles in Photography and Film: When a camera is positioned at a 90° angle to the ground (looking straight up or down), calculations involving sec 90° become undefined. This helps photographers and filmmakers determine extreme angles that may not be practically achievable and adjust their shots accordingly.
• Roller Coaster Design: Engineers designing roller coasters use trigonometry to model steep inclines and vertical drops. When a track reaches a 90° incline, sec 90° is undefined, meaning the track is perfectly vertical. This helps designers understand limitations in real-world physics and ensure safe, realistic ride experiences.