Sec 60°: Definition, Formula & Real-World Applications

Discover the formula, derivation, and real-world applications of Sec 60° in physics, engineering, and daily life. Learn how to solve problems step-by-step.

Sec 60°: Definition, Formula & Real-World Applications

Consider a ladder making an angle of 60° with the ground and leaning against a wall. How can we find the base distance between the ladder and the wall if we know the height of the ladder? This idea, specifically secant, is important to trigonometry. Sec 60° is a basic number that is utilized in everything from engineering to physics computations.

In this article, we’ll explore:

• What is Sec 60°?
• What is the value of Sec 60°?
• Where is Sec 60° applied in real life?

Secant

Secant, represented as Sec, is the reciprocal of the cosine function. Mathematically, for any angle θ in a right triangle:

Sec θ = 1 / cos θ = Hypotenuse / Adjacent side

What is Sec 60°?

The value of the cos trigonometric function for an angle of 60° between the hypotenuse and the adjacent is known as Sec 60°. The formal notation for the secant of a sixty-degree angle is sec 60°.

Value of Sec 60°

We know that Sec 60° is the reciprocal of cos 60°.

Cos 60° = Adjacent / Hypotenuse

Cos 60° = Adjacent / (2 × Adjacent) = 1/2

Sec 60° = 1 / cos 60° = 1 / (1/2) = 2

Therefore, the value of Sec 60° is 2.

Value of Sec 60° in Radians

We know that:

θ in radians = π / 180° × θ in degrees = 60° × (π / 180°) = π / 3

Sec 60° = Sec(π/3) = 2

Value of Sec 60° Using a Unit Circle

On a unit circle, the hypotenuse is always 1 since the radius of the unit circle is 1.

Cos θ = Adjacent / Hypotenuse → Cos θ = x-coordinate of point P

As θ approaches 60°, x-coordinate becomes 1/2 = 0.5

Therefore, Sec 60° = 1 / 0.5 = 2

Value of Sec 60° Using Trigonometric Identities

• Sec 60° = 1 / √(1 − sin²60°) = 2
• Sec 60° = 1 / cos 60° = 1 / 0.5 = 2
• Sec 60° = √(1 + tan²60°)
• Sec 60° = √(cot²60° + 1) / cot 60° = 2
• Sec 60° = cosec 60° / √(cosec²60° − 1) = 2

Examples to Make It Easy

Example 1:

Find the value of cos 60° + sec 60°.

Solution: 1/2 + 2 = 5/2

Example 2:

Solve cos²60° + sec²60°.

Solution: (1/2)² + (2)² = 1/4 + 4 = 17/4

Example 3:

A ladder makes an angle of 60° with the ground and is leaning against a wall. If the length of the ladder is 5 m, find the base distance between the ladder and the wall.

Sec θ = Hypotenuse / Adjacent side = Length of ladder / Base distance

Sec 60° = 5 / Base distance → 2 = 5 / Base distance

Base distance = 5 / 2 = 2.5 m

Therefore, the distance between the ladder and the wall is 2.5 m.

Practice Questions

Test yourself with these problems:

1. Using the value of Sec 60°, find the value of 3 − 2 sec²60°.
2. Find the value of 4Sec 0° if the value of 2sec 0° is 1/2.
3. The arm of a crane makes an angle of 60° with a beam. If the length of the arm of the crane is 8 m, find the horizontal distance between the crane’s arm and the beam.

Real-Life Applications of Sec 60°

• Civil engineering: Used in analysis of forces that act on beams.
• Aerospace and aviation: Used in analysis of approach angles and altitudes.
• Astronomy: Used in telescopes to find the angle of celestial bodies.
• Construction: Used in construction of roads to establish ramps, highway slopes, and angles.

Sec 60° is a straightforward yet effective idea with many practical uses. Knowing cosine makes computations simpler and more effective, whether you're analyzing motion, building structures, or tackling physics problems. Gaining proficiency in this basic trigonometric function can be beneficial in a variety of everyday, scientific, and technical contexts.