Sec 60 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:
The opposing side of the angle of interest is the opposite side.
The hypotenuse side, which is the opposite side of the right angle, is the longest side of a right triangle.
The adjacent side is the remaining side of a triangle, and it is formed by both the angle of interest and the right angle.
The secant function is the reciprocal of the cosine function, and the sec function of an angle is defined as the ratio of the hypotenuse side to the adjacent side, with the formula being
Sec θ = 1 / cos θ
Now, since
Cos θ = Adjacent Side / Hypotenuse Side
Hence,
Sec θ = Hypotenuse Side / Adjacent Side
Sec 60o - Derivation
To derive the value of sec 60 degrees, let us consider an equilateral triangle ABC. As we know each angle of an equilateral triangle is 60°, accordingly, ∠A = ∠B = ∠C.
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 45o+ sec 60o
Solution:
As we know, tan 45o = 1 , and sec 60o = 2
Hence, substituting the values we get:
1 + 2 = 3
Therefore, the value of tan 45o + sec 60o = 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.
FAQs on Sec 60
1.What are the applications of Trigonometric functions?
Trigonometry and its functions can be used in a lot of different ways in our daily lives.
To measure the distance between landmarks.
It is used in geography.
In astronomy to figure out how far away stars are.
The satellite navigation system helps people find their way.
Trigonometry is a study of how angles, lengths, and heights all work together to make sense of the world around us. It started in the third century BC, and it used ideas from astronomy to study geometry. Earlier it was used only for engineering and science but now it has widened its application to even in the investigation of a crime scene also.
2.What are trigonometric special angles and what is so special about them?
Certain angles give simple and exact trigonometric numbers. These are called trigonometric special angles. It includes a list of the degrees. These are 30o, 45o, and 60o.
It is simple to 'precisely' estimate the trigonometric function for these angles without needing a calculator. These angles have rather crisp values, which gives us a lot of flexibility in solving Maths problems.
To get exact answers about the values of many trigonometric ratios, these values are used.
3.What exactly is the Secant Function Reciprocal?
The secant function is the inverse or reciprocal of the cosine function. The cosine function is defined as the relationship between the adjacent and hypotenuse sides of a right triangle in terms of the angle formed by those sides.
As a result, the secant function can be defined as the relationship between a right triangle's hypotenuse and adjacent sides in terms of the angle created between the hypotenuse and adjacent sides.
4.What are the basic Trigonometric Function Formulas?
Basic Trigonometric Function Formulas:
To find the elements in Trigonometry, there are 6 ratios used. They are termed trigonometric functions. The six trigonometric functions are:
Sine
Cosine
Secant
Cosecant
Tangent
Cotangent
The trigonometric functions and identities are derived using a right-angled triangle as a reference:
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
5. What is the use of the Trigonometry formula and how does it help the students?
Different types of problems can be solved utilising trigonometry formulae in trigonometry.
Apart from Trigonometric ratios, Pythagorean identities, product identities some formulas, such as the sign of ratios in different quadrants, cofunction identities (shifting angles), sum and difference identities, double angle identities, half-angle identities, and so on, are also equally important.
Students in Class 10, 11, and 12 will benefit from learning and understanding these trigonometry maths formulas to excel.
They can use the trigonometry table as well to solve real complex problems.
