We may get the acute angle measurements within a right triangle by using the word inverse to trigonometric ratios. Using sine, cosine, or tangent, you can normally compute the other two sides of a right triangle if you are given an angle and one of its sides. You can determine the angle measure using inverse trig ratios if you have two sides. In this article, you will know about the inverse trigonometry tan, about the meaning of tangent and chords in a circle.

Description 

You’re undoubtedly acquainted with the term inverse. If one side length and an angle measure are known, you’ve learned how to apply the trigonometric functions sin(x), cos(x), and tan(x) to get a side length of a right triangle which is not known.

When you learn how to conduct an operation in mathematics, you also learn how to “undo” it. For instance, you may recall that addition and subtraction are inverse operations. Inverse operations include multiplication and division. To solve equations and inequalities in algebra, you make use of inverse operations

The inverse trigonometric functions like sin−1(x),cos−1(x), and tan−1(x) are used to know about the unknown angle of a right-angled triangle when two sides are known.

What is the meaning of tangent?

Well, there are various meanings of a tangent, but in mathematics, we’ve so far come across two meanings in the context of circles and trigonometry ratios.

In Circles, a line that is tangent is known by: 

As the variable point approaches the fixed point, a straight line that represents the limiting position of a secant of a curve via a fixed point and a variable point on the curve.

In Trigonometry, a tangent is known by:

When an acute angle is regarded as part of a right triangle, the trigonometric function is the ratio between the leg opposite the angle and the leg adjacent, for all real integers where the cosine is not equal to zero, a trigonometric function that is equal to the sine divided by the cosine and is precisely equal to the tangent of an angle of measure in radians.

Inverse Functions 

Inverses exist for all mathematical functions. A person can use an inverse function to accomplish the opposite action of the original function. Subtraction is the inverse of addition since it reverses what happened in the additional issue. 

Operation: 

6 is equal to 4 + 2 

Reverse: 

6 minus 2 equals 4 

The division is the inverse of multiplication, squared is the inverse of a square, and the inverse of the tangent function is the inverse tangent function (tan-1).

Meaning of Inverse Tangent 

The inverse tangent is the tangent function’s polar opposite. When you know the sides of a right triangle but not the angle measure, this tool comes in handy. Calculating the inverse tangent is as straightforward as using a scientific calculator. 

The inverse tangent button is located just above the tangent button, and you’ll almost certainly have to hit the inverse key to access it, making it easy to remember.

Chords of a Circle

A circle’s chord is the line segment that links any two points on the circumference of the circle. The diameter of a circle is defined as the longest chord that runs through the centre of the circle.

A circle’s chord is a line segment that links two points on the circumference of the circle. The chord is one of the many line segments that may be drawn in a circle, with its ends on the circumference. A chord that goes through the centre of the circle is also known as a diameter.

Some properties of chords of a circle:

A few key features of a circle’s chords are listed below:

  • The chord is bisected by the perpendicular to a chord drawn from the circle’s centre. 
  • The chords of a circle that is at an equal distance from the centre of a circle are the same.
  • The only circle that goes through three collinear points is the one and only circle. 
  • When a circle chord is drawn, it divides the circle into two sections, which are referred to as the major and minor portions of the circle. 
  • A secant is formed when a chord is prolonged indefinitely on both sides.

Are there any formulas for chords of a circle?

There are two fundamental formulas for calculating the length of a circle’s chord: 

  • Using the perpendicular distance from the centre, the chord length is 2(r2 d2). Let’s look at this formula’s proof and derivation. The hypotenuse of the triangle produced by the circle shown below is radius ‘r.’ One of the legs of the right triangle is the perpendicular bisector ‘d.’ We know that the chord is bisected by the perpendicular bisector from the circle’s centre to the chord. As a result, the other leg of the right triangle is formed by half of the chord. According to Pythagoras’ theorem,

(1/2 chord)2 + d2 = r2, resulting in 1/2 of Chord length = √(r2 − d2).

Thus, chord length = 2 × √(r2 − d2).

  • Using trigonometry, chord length = 2 r sin(θ/2); where ‘r’ is the radius of the circle and ” is the angle subtended by the chord at its centre. The central angle ” subtended by the chord AB and ‘r’ as the radius of the circle may be seen in the following circle.

Some important notes about the chords of a circle 

  • The chord is bisected at 90 degrees by the radius of a circle. 
  • An isosceles triangle is formed when two radii link the two ends of a chord. 
  • The diameter is the longest chord of the circle.

Conclusion 

Well, there are various meanings of a tangent, but in mathematics, we’ve so far come across two purposes in the context of circles and trigonometry ratios. The inverse tangent is the tangent function’s polar opposite. When you know the sides of a right triangle but not the angle measure, this tool comes in handy. The chord of a circle is the line segment that connects any two locations on the circle’s circumference. The diameter of a circle is defined as the longest chord that runs through the centre of the circle.