The tangent of a circle is known as a line touching circles or an ellipse at only one point. Imagine, when a line touches the curve at P, then this point “P” is known as the point of tangency. In differential geometry, the tangent equation can be found using the following methods:
So, we know that finding the gradient of the curve is the gradient of the tangent to the curve at any specified point given on the curve. Hence, the tangent equation of the curve y = f(x) is:
to find the derivative of gradient function through the rules of differentiation.
To find the gradient of the tangent, replace the x- coordinate of the given point in the derivative given.
In the straight-line equation’s slope -point formula, replace the gradient of the tangent and given coordinate point to find out the tangent equation.
Tangent of a Circle Definition
A circle is also known as a curve. It is also a closed two-dimensional shape. It is to be observed that the radius of the circle or the line joining the centre O to the point of tangency or the radius of the circle and tangent line are always perpendicular to each other, i.e. OP is perpendicular to XY as shown in the below figure.
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Here “XY” is the tangent of a circle given, and “OP” is the point of tangency and the tangent radius and the point “O” represents the centre of the circle.
Thus, the radius and the tangent to a circle are related to each other, tangent to a circle formula that can be well explained using the tangent theorem.
Tangent Meaning in Trigonometry
The tangent of an angle is called the ratio of the length of measure of the opposite side to the length of the adjacent side's measure. Hence, it is regarded as the ratio of sine and cosine function of an acute angle; however, the value of cosine function should not equal to zero. It is regarded as one of the six primary functions in trigonometry.
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The direct common tangent formula is:
Tan P = Opposite Side/Adjacent side
Tangent may be given by using Sine and Cosine as:
Tan P = Sin P / Cos P
The sine of an angle is the length of the measure of the opposite side divided by the length of the hypotenuse side's measure. The cosine of the angle is given by the ratio of the length of the measure of the adjacent side to the ratio of the length of the measure of the hypotenuse side.
So, That is, Sin P = Opposite Side/ Hypotenuse Side
Cos P = Adjacent Side/ Hypotenuse Side
tan P = Opposite Side/ Adjacent Side
In trigonometry, the tangent function will help find the slope of a line between the point representing the intersection, the hypotenuse and the altitude of a right triangle and the origin.
Hence, tangent signifies the slope of some object in Trigonometry and tangent geometry. Now let us see at the most important tangent angle – 30 degrees and its derivation.
Derivation of Value of Tangent 30 Degrees
As per the properties of a right-angle triangle when its acute angle equals 30⁰, then the length of the hypotenuse is double the length of the opposite side. The length of the adjacent side is 3√2 times to the length of the measure of the hypotenuse side.
Hence,
Length of Hypotenuse = 2×Length of the measure of the opposite side
Length of Adjacent side= √3/2 × Length of Hypotenuse
Length of Adjacent side= √3/2 × (2×Length of Opposite side)
Length of Adjacent side= (√3/2×2) ×Length of Opposite side
Length of Adjacent side=√3 × Length of Opposite side
1√3=Length of opposite side/length of the adjacent side
Since the ratio is tan30⁰,
tan30⁰ = 1/√3
Similarly, we can find the values of other angles like 45, 60 using this property of right-angled triangles.
Applications of Tangents in Science and Technology
Tangent has a wide range of use in science and technology as it is the function of sine and cosine. Some of the areas that use trigonometric functions are the Artificial Neural Networks, visualisations, behaviour of elementary particles, and waves like sound waves, electromagnetic waves.
FAQs on Tangents in Geometry
1. What is a tangent in geometry, and how is it different from a secant?
A tangent is a straight line that touches a curve, such as a circle, at exactly one point. This specific point is known as the point of tangency. The line does not cross into the interior of the circle. In contrast, a secant is a line that intersects a circle at two distinct points, passing through its interior.
2. What is the relationship between a tangent and the radius of a circle at the point of tangency?
A fundamental theorem in geometry, as per the CBSE syllabus, states that the tangent at any point of a circle is perpendicular (forms a 90° angle) to the radius that is drawn to that same point of tangency. This property is crucial for solving many problems involving circles and tangents.
3. What is the main property of tangents drawn from an external point to a circle?
According to a key theorem, the lengths of the two tangents drawn from the same external point to a single circle are always equal. For instance, if two tangents from an external point P touch the circle at points A and B, the length of the segment PA will be exactly equal to the length of the segment PB.
4. Why can there be only one tangent at any single point on a circle?
A tangent is defined by its property of touching the circle at a single point without entering it. If you were to draw a line at this point and tilt it even slightly, it would immediately cut through the circle at a second, nearby point, which would make it a secant. The only orientation that avoids this is the line that is perfectly perpendicular to the radius at that point, which guarantees there is only a single point of contact.
5. How many tangents can be drawn to a circle from a single point?
The number of tangents that can be drawn depends entirely on the location of the point relative to the circle:
- If the point is inside the circle, zero tangents can be drawn.
- If the point is on the circle, exactly one tangent can be drawn.
- If the point is outside the circle, exactly two tangents can be drawn.
6. Is the 'tangent' in geometry the same as the 'tangent' (tan) in trigonometry?
No, they are different concepts that are related but not identical. The geometric tangent is a line that touches a curve at one point. The trigonometric tangent (tan) is a function that represents the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. The concepts are linked because the length of a geometric tangent segment can sometimes be calculated using the trigonometric tangent function in specific geometric setups.
7. What does the 'length of a tangent' refer to?
The 'length of a tangent' specifically refers to the length of the line segment measured from an external point to the point of tangency on the circle. If a tangent is drawn from an external point P to a point of contact T on the circle, the 'length of the tangent' is the distance of the segment PT.
8. What happens to the angle between two tangents from an external point as the point moves further from the circle?
As the external point moves further away from the circle's centre, the two tangent lines become more parallel to each other. Consequently, the angle formed between them at the external point decreases. If the point were to move infinitely far away, the tangents would become effectively parallel, and the angle between them would approach zero degrees.
9. How are the properties of tangents used in real-world examples?
The principles of tangents are applied in many practical fields. For example:
- In engineering, they are fundamental to designing belt and pulley systems, where a belt runs tangent to the wheels to transfer power.
- In physics, the instantaneous velocity of an object in uniform circular motion is always directed tangent to the circular path.
- In optics, tangents help describe how light rays reflect off or refract through curved lenses and mirrors.
