How to Calculate Sec 60 and Apply It in Maths Problems
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: Meaning, Derivation & Applications
1. What is the exact value of sec 60 degrees?
The exact value of sec 60° is 2. The secant function is one of the six fundamental trigonometric ratios and is the reciprocal of the cosine function. For an angle of 60 degrees, this value is a whole number, making it a commonly used value in mathematical problems.
2. How is the value of sec 60° related to cos 60°?
The secant function is the reciprocal of the cosine function. This relationship is defined by the formula sec(θ) = 1 / cos(θ). To find the value of sec 60°, you first need the value of cos 60°, which is 1/2. By applying the reciprocal identity, we get:
sec 60° = 1 / cos 60° = 1 / (1/2) = 2. This inverse relationship is fundamental to understanding trigonometric functions.
3. How can you find the value of sec 60° using a geometrical proof?
You can derive the value of sec 60° geometrically using an equilateral triangle. Here are the steps:
Start with an equilateral triangle ABC with each side measuring 2 units and each angle being 60°.
Draw a perpendicular line AD from vertex A to the base BC. This line bisects the base BC at D and the angle at A.
Now consider the right-angled triangle ADB. In this triangle, the angle B is 60°, the hypotenuse AB is 2 units, and the adjacent side BD is 1 unit.
The secant of an angle in a right-angled triangle is the ratio of the length of the hypotenuse to the length of the adjacent side.
Therefore, sec 60° = Hypotenuse / Adjacent = AB / BD = 2 / 1 = 2.
4. Why are standard angles like 30°, 45°, and 60° considered important in trigonometry?
Standard angles like 30°, 45°, and 60° are important because their trigonometric values (sin, cos, tan, etc.) are exact and can be expressed as simple fractions or surds without needing a calculator. This allows for:
Precise Calculations: They are used to solve problems in geometry, physics, and engineering with exact answers.
Foundation for Learning: They serve as fundamental benchmarks for understanding the behaviour of trigonometric functions.
Problem Solving: These values frequently appear in academic problems and competitive exams, and knowing them from the trigonometry table speeds up solving complex questions.
5. What is the value of sec 60° in terms of tan 60°?
The relationship between secant and tangent is given by the Pythagorean identity: sec²(θ) = 1 + tan²(θ). To find sec 60° using tan 60°, you can follow these steps:
The value of tan 60° is √3.
Substitute this value into the identity: sec²(60°) = 1 + (√3)².
Calculate the square: sec²(60°) = 1 + 3 = 4.
Take the square root: sec(60°) = √4 = 2 (we take the positive root as 60° is in the first quadrant).
6. How does the value of sec(θ) change as the angle θ increases from 60° towards 90°?
As the angle θ increases from 60° and gets closer to 90°, the value of cos(θ) decreases and gets closer to 0. Since sec(θ) = 1 / cos(θ), dividing 1 by a smaller and smaller positive number results in a larger and larger value. Therefore, the value of sec(θ) increases rapidly and approaches infinity as θ approaches 90°. This is why sec 90° is considered undefined.
7. Where could the value of sec 60° be used in a real-world problem?
The value of sec 60° can be used in problems involving heights, distances, and engineering. For example, consider a ladder leaning against a wall. If you know the angle the ladder makes with the ground is 60° and the distance from the base of the wall to the foot of the ladder is 5 metres, you can find the length of the ladder.
Using the formula sec(θ) = Hypotenuse / Adjacent:
sec 60° = Length of Ladder / 5 metres
2 = Length of Ladder / 5
Length of Ladder = 2 × 5 = 10 metres.
This is a typical problem in the application of trigonometry.
