Introduction to Trigonometric Functions
Trigonometric functions like sine, cosine, tangent, cosecant, secant and cotangent form a key part of the class 11 maths curriculum as per NCERT. These functions establish relationship between angles and sides of a right-angled triangle and are widely used in mathematical and real-world applications. In this blog, we will explore the basics of trigonometric functions, their representations, values and illustrations from NCERT solutions for class 11 maths.
What are Trigonometric Functions?
Trigonometric functions provide the mathematical relationship between the angles and sides of a right triangle. For a right triangle with hypotenuse h and opposite and adjacent sides as a and b respectively, the six trigonometric functions are defined as:
sin θ = opposite/hypotenuse = a/h cos θ = adjacent/hypotenuse = b/h tan θ = opposite/adjacent = a/b cosec θ = hypotenuse/opposite = h/a
sec θ = hypotenuse/adjacent = h/b cot θ = adjacent/opposite = b/a
Where θ is the angle opposite side a in the triangle.
Representations of Trigonometric Functions
Trigonometric functions can be represented in three ways:
As ratios of sides of a right triangle
On the unit circle
As coordinate points on the trigonometric graph
The NCERT maths textbook provides detailed explanations on all three representations.
Values of Trigonometric Ratios
The NCERT solutions provide exhaustive tables of values of trigonometric ratios of standard angles from 0° to 360°. These include sine, cosine, tangent, cosecant, secant and cotangent values for angles that are multiples of 30° and 45°. Memorizing these values is important for solving trigonometry problems faster.
Illustrations from NCERT Class 11 Maths
Let us take a couple of examples from the NCERT solutions for class 11 maths chapter 8 on trigonometry applications to illustrate their usage.
Example 1: Find sin 75°, cos 75° and tan 75°
Solution: We know sin 75° = sin (60° + 15°) = sin 60°cos 15° + cos 60°sin 15° = (√3/2)(√2/2) + (1/2) = (√6 + 1)/2
Similarly, cos 75° = (√3/2)(√2/2) - (1/2) = (√6 - 1)/2 tan 75° = sin 75°/cos 75° = (√6 + 1)/(√6 - 1)
Example 2: If tan A = 3/4 for an acute angle A, find sin A and cos A.
Solution: Using tan A = sin A/cos A, we get: sin A = 3/5, cos A = 4/5
Applications of Trigonometry
Trigonometry has many applications in fields like astronomy, physics, engineering, navigation and surveying. Some uses are determining heights and distances, calculating wave properties and analysing periodic motions.
This highlights the importance of gaining proficiency in trigonometric concepts and their applications from the NCERT textbook and solutions for class 11. Regular practice with the detailed explanations given in the NCERT maths solutions is key for mastering the topic.
In summary, trigonometry is an essential branch of mathematics with immense utility in real world scenarios. A strong understanding of trigonometric functions as detailed in the NCERT class 11 maths curriculum will enable students to excel in higher level maths and science subjects and associated career domains.