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Sec 30 

 

Sine, cosine, and tangent are the three essential trigonometric functions. The other three trigonometric functions are derived from these three functions, namely secant, cosecant, and cotangent, commonly known as a sec, cosec, and cot.

Secant is also the reciprocal of the cosine function. It is the ratio of the length of the hypotenuse to the length of the adjacent side of the angle considered in a right-angled triangle.

Secant = hypotenuse / base of the angle considered

Value of sec 30

From the given diagram, we can see sec alpha = hypotenuse / base of the right-angled triangle.

This gives, sec ∠α = h/b

Also, sec ∠α = 1/cos ∠α

Therefore, sec 30 = 1/cos 30

We know the value of cos 30 = √3/2

Hence, the value of sec 30 = 2/√3

Example

Find the value of (2 sec 30° + 2 cos 60°).

Solution

We know, sec 30° = 2/√3 and cos 60° = 1/2

Therefore, 2 sec 30° + 2 cos 60° = 2 × 2/√3 + 2 × ½

= 4/√3 + 1

2 sec 30° + 2 cos 60° = 4/√3 + 1

Example

Find the value of Cos 30° x Sec 30° - Sec² 30°.

Solution

Cos 30° x Sec 30° - Sec² 30° = 3/√2 × 2/√3 − (3/√2)²

= 1 – ¾ = ¼

Common values of various trigonometric functions

  30° 45° 60° 90°
Sin 0 1/2 1/√2 √3/2 1
Cos 1 √3/2 1/√2 1/2 0
Tan 0 1/√3 1 √3
Sec 1 2/√3 √2 2
Cosec 2 √2 2/√3 1
Cot √3 1 1/√3 0

Important formulas of secant angles-

sec 2 y = 1 + tan 2 y

sec(90°–x) = x

sec(2x) = x/(2–x)

Inverse of a secant function

The inverse of a secant function is known as the arcsec function. Whenever we see arcsec A written, we can infer the angle whose secant is A.

arcsec x = sec-1 x

sec 60 = 2.000   Means: The secant of 60 degrees is 2.000

arcsec 2.0 = 60   Means: The angle whose secant is 2.0 is 60 degrees.

Real-life uses of trigonometry

  1. The heights of buildings and mountains are calculated with the help of trigonometry. From the basic formulas and identities, surveyors find an approximate value of various heights of sculptures and land levels.
  2. Trigonometry is used to make video games. How much a character must jump to overcome the obstacle, what should be its height and the enemy’s height, all is found out with the help of trigonometry.
  3. Trigonometry is used to do aerial surveys. It is used to capture different geographical and topographical regions to construct maps and plot them on paper.
  4. In physics, trigonometry is the most widely used topic. It is used to find the components of vectors, calculate the sum of strength of fields, model the mechanics of waves and oscillations (both electromagnetic and physical), and use dot and cross products. It is also widely used in projectile motions.
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