Trigonometric Substitution
Question 1: What is trigonometric substitution?
Answer:
Trigonometric functions are substituted in order to compute results for other expressions. This is known as a trigonometric substitution. It is used for the evaluation of regular functions and integrals can also be simplified using substitution methods when the expressions are formed using radicals. Also, the antiderivative can be easily solved before applying the boundaries of integration, and therefore, the process is amplified using trigonometric substitution.
Question 2: What is a conic section in real life?
Answer:
There are a large number of conic section examples in real life for example, if we consider the Sun as the focus point, then the path or the trajectory of the planets rotating around the sun forms an ellipse. And the other example is the case of parabolic mirrors, which capture the light beam at the focus of a parabola.
Question 3: What is the formula for the trigonometric equation of a circle?
Answer:
To find the formula for the trigonometric equation of the circle.
We have to,
Assume,
x = acosθ,
y = asinθ
x2 + y2 = a2cos2θ + a2sin2θ
x2 + y2 = a2( cos2θ + sin2θ)
x2 + y2 = a2 (by using cos2θ + sin2θ=1)
x2 + y2 = a2
Question 4: What are trigonometric identities?
Answer:
The trigonometric identities are the expressions that you make use of the trigonometric functions. These trigonometric identities can be considered as the laws in mathematics and therefore, these values of identities hold true for all sets of values provided in them. For instance, trigonometric identities are considered to be appropriate for a right-angled triangle wherein there are six trigonometric ratios that are used to characterize any right-angle triangle. Therefore, irrespective of the side lengths of the right angle triangle, the trigonometric ratio ratios as well as identities are satisfied by all of them. An important feature of the trigonometric identity is that it is made up of using the side length as well as the angles between the sides and then computed.
Question 5: Calculate tan 45º from the unit circle by using the values of sin and cos.
Answer:
Here we have to find the value of tan 45º
As we know that
tanθ = sinθ/cosθ
So according to the problem
tan 45° = sin 45°/cos 45°
As we know that
sin 45° = 1/√2
cos 45° = 1/√2
Thus,
tan 45° = sin 45°/cos 45°
= {1/√2}/1/√2}
= 1
Therefore,
tan 45° = 1
Use Trigonometric Substitution and Trigonometric Identity to determine the Circumference of a Circle of Radius 3.
When any two-dimensional or three-dimensional plane is being intersected and cut then there are curves obtained, which are known as conic sections. In the case of a cone, there are two identical conical shapes, which are also known as capes. The shape of these conical shapes in the cone depends upon the angle that is opting in as a result of the cut between the plane and the cone. Various shapes can be obtained depending upon the difference in the curve sizes which are circle, hyperbola, ellipse, and parabola we can study these shapes in detail, wherein
- The ellipse is one of the conic sections, which is formed as a result of the intersection of a plane with the cone at an angle.
- The circle is considered to be a conic section where the intersection happens parallel to the cone base and it can be considered to be a special type of ellipse.
- Also, hyperbola formation occurs when the plane is parallel to the cone axis and the intersection happens with both the names of the double cone.
- Lastly, we obtain a parabola when the intersecting plane cuts the surface of the cone at an angle.
The following diagram represents the various types of conic sections that are available depending on the difference in the cuts between the intersecting surface and the cone and its necks.
In this article, we will focus on the conical section of the circle. First, the circle has the following properties,