Sec 30 Degree – Calculation, Derivation, Solved Examples & FAQs

The value of sec 30 degrees is approximately 1.1547005. When expressed in radians, sec 30 degrees is written as sec (30° × 180°), which simplifies to sec 180° or sec (0.5235987) . In this article, we’ll discuss various methods to find the value of sec 30 degrees, solved examples related to sec 30 and more.

Sec 30° = 2√3

Sec 30° in decimal = 1.1547005. . .

Sec (-30 degrees) = 1.1547005. . . or 2/√3

Sec 30° in radians: sec (π/6) or sec (0.5235987 . . .)

Also Check: Curve

Sec 30 Value

The value of sec 30 degrees in decimal form is approximately 1.154700538. This trigonometric function can also be represented in radians, where 30 degrees is equivalent to about 0.52359 radians.

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Converting Degrees to Radians

To convert an angle from degrees to radians, use the formula:

in radians = in degrees (180°)

Applying this to 30 degrees:

30° = 30° (180°) = 6 0.52359

Therefore, sec 30 degrees can be expressed as:

Sec 30° = sec(0.52359) = 2√3 1.1547005

Sec 30 Degrees

The angle of 30° lies in the first quadrant, between 0° and 90°, where the secant function is positive. Therefore, the value of sec 30° is 2√3 or is approximately equal to 1.1547005.

Periodicity and Properties of the Secant Function

Since the secant function is periodic, sec 30° can be represented as:

Sec 30° = Sec (30° + n 360°), n Z

This means, Sec 30° = Sec 390° = Sec 750° and so on.

Even Function Property

Additionally, secant is an even function, which implies that: Sec(- 30°) = Sec 30°

Also Check: Area of Quadrilateral

Methods to Find the Value of Sec 30 Degrees

The secant function is positive in the first quadrant, and the value of sec 30° is approximately 1.15470. There are two primary methods to determine the value of sec 30 degrees:

1. Using Trigonometric Functions

Sec 30 degrees can be found using the relationship between secant and cosine:

Sec 30° = 1cos 30°

We all know that cos 30° = 32

Therefore, Sec 30° = 1cos 30° = 132 = 23

2. Using the Unit Circle

On the unit circle, 30 degrees corresponds to the point (cos 30° , sin 30° ). Since secant is the reciprocal of cosine, the secant value can be derived from the x-coordinate:

Sec 30° = 1cos 30° = 132 = 23

To find sec 30° using the unit circle, follow these steps:

• Start by rotating a radius ‘r’ anticlockwise to form a 30° angle with the positive x-axis.
• The point of intersection between the unit circle and line ‘r’ at 30° is approximately (0.866, 0.5), where 0.866 is the x-coordinate.
• Now calculate the value of Sec 30° . The secant of 30 degrees is the reciprocal of the x-coordinate:

Sec 30° = 1cos 30° = 132 = 23 =1.1547

Therefore the value of Sec 30° is 1.1547.

Also Check: Average

Sec 30 Trignometric Formulas

Using trigonometric formulas, sec 30 degrees can be expressed in various forms:

1. Sec 30° = 11 – sin230°
2. Sec 30° = 1 + tan230°
3. Sec 30° = 1 + cot230°cot 30°
4. Sec 30° = cosec 30cosec230° – 1
5. Sec 30° = 1cos 30°

Note: Since 30° is in the first quadrant, where all trigonometric functions are positive, the final value of sec 30° will be positive.

Trigonometric Identities Involving Sec 30°

Using various trigonometric identities, sec 30° can also be represented as:

1. – Sec(180 – 30°) = -Sec 150°
2. – Sec(180 + 30°) = -Sec 210°
3. cosec(90 + 30°) = cosec 120°
4. cosec(90 – 30°) = cosec 60°

Also Check: Circumstance of Circle

Trigonometric Identities

Below mentioned are a few basic trignometric identities.

1. sin = 1cosec ; cosec = 1sin
2. cos = 1sec ; sec = 1cos
3. tan = 1cot ; cot = 1tan
4. cos2 + sin2 = 1
5. sec2 – tan2 = 1
6. cosec2 – cot2 = 1
7. sin (90°- θ) = cos θ
8. cos (90°- θ) = sin θ
9. tan (90°- θ) = cot θ
10. cosec (90°- θ) = sec θ
11. sec (90°- θ) = cosec θ
12. cot (90°- θ) = tan θ
13. sin (180°- θ) = sinθ
14. cos (180°- θ) = -cos θ
15. tan (180°- θ) = -tan θ
16. cosec (180°- θ) = cosec θ
17. sec (180°- θ)= -sec θ
18. cot (180°- θ) = -cot θ
19. sin (A+B) = sin A cos B + cos A sin B
20. sin (A-B) = sin A cos B – cos A sin B
21. cos (A+B) = cos A cos B – sin A sin B
22. cos (A-B) = cos A cos B + sin A sin B
23. tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
24. tan (A-B) = (tan A – tan B)/(1 + tan A tan B)
25. sin (x + 2π) = sin (x)
26. cos (x + 2π) = cos (x)
27. tan (x + π) = tan (x)

Trigonometric Values

Below given are the trignometric values at different angles.

0°	30°	45°	60°	90°
Sin	0	12	12	32	1
Cos	1	32	12	12	0
Tan	0	13	1	3
Sec	1	23	2	2
Cosec	2	2	23	1
Cot	3	1	13	0

Sec 30 Solved Examples

1. Find the value of Sec 390°

Solution: We know that the secant function is periodic with a period of 360°.

Therefore, Sec 390° = Sec (390° – 360°) = Sec 30° = 23 =1.1547

Therefore the value of Sec 390° is 23 =1.1547.

2. Find the value of Sec 750°

Solution: We know that the secant function is periodic with a period of 360°.

Therefore, Sec 750° = Sec (750° – 2 360°) = Sec 30° = 23 =1.1547

Therefore the value of Sec 750° is 23 =1.1547.

Sec 30 Practice Questions

1. Find the value of Sec 570°

2. Evaluate the value of Sec 1110°

3. Find the value of Sec 1170°

4. Solve for the value of Sec 1410°

5. Evaluate the value of Sec 1470°

Sec 30 FAQs

What is the exact value of Sec 30°?

The exact value of Sec 30° is 2√3 or 1.15470.

Why is Sec 30° positive?

As we know 30° lies in the first quadrant, where all trigonometric functions are positive. Therefore, Sec 30° is positive.

What is the decimal approximation of Sec 30°?

The decimal approximation of Sec 30° is 1.1547.