Sec 60 in Trigonometry Definition and Exact Value

The word Trigonometry is derived from two roots, i.e., trignon means "triangle", and metric means "measure". The study of trigonometry is based on the study of measurement of the triangle. What can we measure in a triangle? We can simply measure the length of the sides, angles of a triangle, and the area included in a given triangle.

The trigonometric functions are usually thought of as angle functions that deal with triangles and connect the angle of a triangle to the length of the triangles. Trigonometric functions are used in many fields, like engineering and architecture. 

We can measure the angles and length of the sides of a right-angled triangle with the help of trigonometric ratios. Sine, cosine, tangent, secant, cosecant, and cotangent are six trigonometric ratios that represent the relation between the angles, and the length of the sides of a right-angled triangle. The ratio of the three sides of a right-angled triangle in terms of any of its acute angles is considered as the trigonometry ratio of that specific angle.

The sine function, the cosine function, and the tangent function are the three most common trigonometric ratios. The inverse function is called the cosecant function (cos), the secant function (sec), or the cotangent function (cot). 

In this article, we will briefly discuss sec 60, and how to derive the secant 60 value.

Sec 60 Degrees - Value

As we know sec 60 degree or sec 60 value is 2. Let us first know the importance of the secant function in trigonometry, before discussing how the sec 60 value is derived geometrically.

To figure out the function of an acute angle, think about a right triangle ABC with the angle of interest and the sides of a triangle. The sides of the triangle are specified as below:

Construction: Draw a perpendicular line AD from A to the side BC.

In ΔABD and ΔACD

∠B = ∠C.

AB = AC ( As we know all the sides of an equilateral triangle are equal)

AD = AC( Common side)

Hence, by the Angle-Side Angle theorem, we can say that 

ΔABD ≅ ΔACD

So, BD = DC, and ∠BAD = ∠ CAD ( By CPCT)

It is seen that ABD is a right triangle, right angled at D with

∠BAD = 30°, and ∠ ABD = 60°

To determine the trigonometric ratios, we need to determine the length of the sides of the triangle. So, let us assume the side AB = 2a, and BD = BC/2 = a.

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

Cos θ = Adjacent Side/ Hypotenuse Side

Cos 60° = BD/AB

Cos 60° = a/2a = ½

As we know, the secant function is the inverse function of the cosine function, it becomes:

Sec 60° = 1/ cos 60°

Sec 60° = 1/ (½) = 2

Hence, the value of Sec 60° = 2

Similarly, we can derive the other degrees of sec value like 0°, 30,° 45°, 90°, 180°, 270°, and 360°.

Apart from the basic derivation above, there are two other techniques for determining the value of Sec 60 Degrees.

In the first quadrant, the secant function is positive. Sec 60° is denoted by the number 2. We may determine the value of sec 60 degrees as follows:

• Through the Unit Circle method

• Through Trigonometric Functions

Using the Unit Circle, 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 sec of 60 degrees.

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

Using the Trigonometric Functions, Find theValue of Sec 60 Degrees:

± 1/√(1 - sin²(60°))

± √(1 + tan²(60°))

± √(1 + cot²(60°))/cot 60°

± cosec 60°/√(cosec²(60°) - 1)

1/cos 60°

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

To represent sec 60°, we can utilise trigonometric identities like,

-sec(180° - 60°) = -sec 120°

-sec(180° + 60°) = -sec 240°

cosec(90° + 60°) = cosec 150°

cosec(90° - 60°) = cosec 30°

Solved Examples

1.What is the value of cos 60° + sec 60°.

Solution:

Since, cos 60° = ½ and sec 60° = 2

So,

cos 60° + sec 60° = (½) + 2 

= (1 + 4)/2

= 5/2

2.Calculate the value of sec 300°

Solution:

Sec 300° = Sec (360 - 60)°

= Sec 60°, As we know, sec (360° - θ ) = sec θ.

Hence value of sec 300°is 2

3.Find the value of tan 45^o+ sec 60^o

Solution: 

As we know, tan 45^o = 1 , and sec 60^o = 2

Hence, substituting the values we get:

1 + 2 = 3

Therefore, the value of  tan 45^o + sec 60^o = 3.

Conclusion: 

Trigonometry is an important function of applied mathematics and a tough discipline at that. Conceptual clarity is of utmost importance here. Students can polish their skills by practising via solved examples.