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Secant Formula

Trigonometry is an important branch of Mathematics. It defines six trigonometric ratios based on a right-angled triangle. The three well known and common trigonometric ratios are sine, cosine, and tangent. However, there an additional three ratios which are secant, cosecant, and cotangent. This article will explain the concept of secant with secant formula and examples. Let us begin!

What is Secant?

Trigonometry mainly deals with the triangles and their angles. It provides popular relationships between the lengths and angles of triangles. Also, it provides many formulas and identities as well as equations, to help us for many simple as well as complex computations in science and maths. There are a large number of applications of these trigonometry based identities and formulas.

There are six ratios which are the core of trigonometry. These are,

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Secant (sec)
  • Cosecant (cosec)
  • Cotangent (cot)

Out of these six trigonometry ratios, three are the basic and the other three are derived ratios. Secant is derived from the cosine ratio. In a right-angled triangle, the secant of any angle will be the ratio of the length of the hypotenuse and the length of the adjacent side. In formulas, it is abbreviated as ‘sec’. It has a period of 2 \pi, similar to sine and cosine.

Secant Formula

Source: en.wikipedia.org

The Formula for Secant

In the right-angled triangle, we have three sides i.e. Hypotenuse, the opposite side (Perpendicular) and Adjacent side (Height). The largest side is the hypotenuse, the side opposite to the angle is opposite and the side where both hypotenuse and opposite rests is the adjacent side.

The length of the hypotenuse, when divided by the length of the adjacent side, will give the secant of the angle in a right triangle. Therefore, its basic formula is:

\(sec X = \frac{Hypotenuse}{Adjacent Side}\)

Also, it is the reciprocal of the cosine value. So,

\(sec X = \frac {1} {cos X}\)

The Pythagorean formula is also there for the secant of the angle, which is as below:

\(Sec ^{2} x – tan ^2{x}\) will be always 1

This equation is very useful. It is similar to the squared relationship between sin X and cos X.

Solved Examples for Secant Formula

Q.1: Find Sec X if Cos x is given as \(\frac{3}{8}\) using a secant formula.

Solution: As we know that  \(Sec X = \frac {1} {Cos X}\)

\(Sec X = \frac{1}{\frac{3}{8}}\)

\(Sec X = \frac {8}{3}.\)

Therefore the value of Sec X will be \(\frac{8}{3}.\)

Q.2: Compute the value of the secant of the angle in a right triangle, having hypotenuse as 5 and adjacent side as 4.

Solution: As we know that, the formula for secant of angle X is:

\(sec X = \frac{Hypotenuse}{Adjacent Side}\)

Thus substituting the values we get,

\(sec X = \frac{5}{4}\)

So, the value of the secant will be \(\frac{5}{4}.\)

Q.3: Determine the value of sec \(\theta\), if \( tan \theta\) is given as 1, in a right-angled triangle.

Solution: The given parameter is,

\(Tan \theta = 1\)

We know the equation as follows:

\(sec ^{2} x – tan ^{2} x = 1\)

i.e. \(sec ^{2} X = 1 + tan^{2} X\)

\(= 1 + 1 ^{2} = 2\)

\(sec X = \sqrt {2}\)

Therefore, the value of secant of the angle will be \(\sqrt {2}.\)

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