Sec 60°: Meaning, Value, Properties, Derivation, Methods

How can we find the Height of a Mountain or distance from its peak? Or How wide is a river? It is possible to calculate the heights of large mountains or the width of a river using trigonometry. The concept of trigonometry states that "if two triangles have the same set of angles, then their sides have the same ratio." Side lengths vary, but side ratios remain constant. Many professions, such as engineering, navigation, and architecture, rely on trigonometric functions. It may be used to calculate distances, find motion routes, and examine waves, among other things. Let’s discuss in detail the value of sec 60 degrees and how the values are derived geometrically.

Key Takeaways: Sine, Cosine, Tangent, Secant, Cosecant, Cotangent, Unit Circle, Trigonometric Functions.

Also read: Isosceles Triangle Theorems

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Meaning of Trigonometry

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Trigonometry where 'Trigon' refers to a triangle, and 'metry' refers to a measurement, is a discipline of mathematics concerned with the connections between triangles' sides and angles, particularly right-angle triangles. In mathematical terms, these connections are represented by a collection of functions known as trigonometric functions. Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent are the six trigonometric functions.

Read more: Trigonometry 

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Value of Sec 60° 

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Sec 60° has a value of 2. In radians, sec 60° is expressed as sec (60° × π/180°), i.e. sec (π/3).

Consider a right triangle ABC with the angle of interest and the triangle's sides to determine the function of an acute angle. The sides of the triangle are specified in the diagram below.

• The opposite side of the angle of interest is the Perpendicular (P).
• The longest side of a right triangle is the Hypotenuse (H), which is the opposite side of the right angle.
• The remaining side of a triangle is created by both the angle of interest and the right angle, and it is called the Base (B).

The reciprocal of the cosine function is the secant function, and the sec function of an angle is defined as the ratio of the hypotenuse side to the neighboring side, with the formula being.

Sec θ = 1/Cosθ

Now, since

Cos θ = Base/Hypotenuse=B/H

Hence,

Sec θ = Hypotenuse/Base = H/B

Read more: Right Triangle

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Derivation of Secant 60°

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Let's look at an equilateral triangle ABC to find the value of sec 60 degrees. An equilateral triangle's angles are all 60 degrees, so ∠A = ∠B = ∠C

From A to the side BC, draw a perpendicular line AD.

In ΔABD and ΔACD

AB = AC (Because all the sides of an equilateral triangle are equal)

∠B = ∠C = 60°

And ∠BAD = ∠CAD = 30°

AD = AD (Common Side)

So, by Side Angle Side Theorem

ΔABD ≅ ΔACD

Now, BD = DC (By C.P.C.T.)

To calculate the trigonometric ratios, we must first establish the length of the sides of a given triangle. So, let's suppose the side

AB=2x, and BD=BC/2=x

Let us consider ΔABD to determine the value of cos 60° in which AB = 2x, and BD = x. Accordingly,

Cos θ = Base/Hypotenuse=B/H

Cos 60° = BD/AB

Cos 60° = x/2x = 1/2

Also we know that

Sec θ = Hypotenuse/Base = H/B

Hence

Sec 60° = 1/ cos 60°

Sec 60° = 1/ (½) = 2

Hence, the value of Sec 60° = 2

Other values of sec degrees, such as 0°, 30°, 45°, 90°, 180°, 270°, and 360°, can be calculated in the same way. In the trigonometry table below, the secant function and various trigonometric ratios are discussed.

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Methods for Finding Secant 60°

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In the first quadrant, the secant function is always positive. Sec 60° is provided as a value of 2. The value of sec 60 degrees can be found by:

• Unit Circle
• Trigonometric Functions

By Unit Circle

Using the unit circle, To find the value of sec 60 degrees,

• Anticlockwise rotate 'r' to produce a 60° angle with the positive x-axis.
• The reciprocal of the x-coordinate (0.5) at the point of intersection (0.5, 0.866) of the unit circle and r is the secant of 60 degrees.

As a result, sec 60° = 1/x = 2 is the value.

Read more: Unit Circle

By Trigonometric Functions

We can represent the secant 60 using trigonometric functions.

± 1/√(1 - sin²(60°))

± √(1 + tan²(60°))

± √(1 + cot²(60°))/cot 60°

± cosec 60°/√(cosec²(60°) - 1)

1/cos 60°

The true value of sec 60° will be positive because 60° is in the first quadrant.

Or,

secθ=1/cosθ (By Identity)

cos 60°=1/2

Hence,

Sec 60°=2

Sec 60° can also be represented as,

-sec(180° - 60°) = -sec 120°

-sec(180° + 60°) = -sec 240°

cosec(90° + 60°) = cosec 150°

cosec(90° - 60°) = cosec 30°

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Facts About Trigonometry

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• Hipparchus, a Greek mathematician, is generally credited with developing trigonometry.
• A 'Shadow Stick,' which casts shadows to monitor the movements of the Sun and thus tell the time, is the most ancient technology found in all early civilizations.
• Trigonometry was developed by the Babylonians and Egyptians using their base 60 number system.
• Trigonometry is used by engineers to determine the angles of sound waves and how to build rooms.

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Some Important Formulas

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In Trigonometry, there are essentially six ratios that are utilized to find the elements. Trigonometric functions are what they're called Sine, cosine, secant, cosecant, tangent, and cotangent.

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side
• The reciprocal identities are given below.
• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• We can use trigonometric formulas to get the sine, cosine, tangent, secant, cosecant, and cotangent values if we know the height and the base side of the right triangle. Trigonometric functions are also used to generate reciprocal trigonometric identities.

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Things to Remember

• The term "trigonometry" is derived from "triangle measure", where 'Trigon' refers to a triangle, and 'metry' refers to measurement.
• There are six trigonometric functions: sine, cosine, tangent, secant, cosecant and cotangent.
• Trigonometry is based on the relationship between the sides and angles of a right triangle.
• The hypotenuse is the side opposite the right angle. Legs are the sides that go parallel to the right angle.
• Secant, cosecant and cotangent are reciprocal functions of sine, cosine, tangent.
• If you know the angle between two lengths, you can use trigonometry to convert them.

Read more: Properties of Triangle