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JEE Important Chapter - Trigonometry

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Last updated date: 23rd Apr 2024
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Introduction of Trigonometry

Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a right-angled triangle. It is one of the most important branches in the history of mathematics. Hipparchus, a Greek mathematician, introduced this concept. We will learn the fundamentals of trigonometry in this article, including trigonometry functions, ratios, the trigonometry table, formulas, and many solved examples.


JEE Main Maths Chapter-wise Solutions 2023-24


Important Topics of Trigonometry

  • Trigonometric Ratios

  • Trigonometry - Formulas, Identities, Functions and Problems


Important Trigonometry Concepts

What is Trigonometry?

The word Trigonometry is clubbed as, 'Trigonon' which means triangle and 'Metron' means to measure. The branch of mathematics known as "trigonometry" studies the relationship between the sides and angles of a right-angle triangle. Using trigonometric formulas, functions, or identities, it is possible to find the missing or unknown angles or sides of a right triangle. The angles in trigonometry can be measured in degrees or radians. 0°, 30°, 45°, 60°, and 90° are some of the most commonly used trigonometric angles in calculations.

Trigonometry is further divided into two subcategories. The following are the two types of trigonometry:

Plane Trigonometry and Spherical trigonometry


Basic Trigonometry

The measurement of angles and problems involving angles are covered in the fundamentals of trigonometry. Trigonometry has three basic functions: sine, cosine, and tangent. Other important trigonometric functions can be derived using these three basic ratios or functions: cotangent, secant, and cosecant. These functions are the foundation for all of the important concepts in trigonometry.


Ratios in trigonometry: Sine, Cosine, and Tangent

The trigonometric functions are the trigonometric ratios of a triangle. The trigonometric functions sine, cosine, and tangent are abbreviated as sin, cos, and tan. Let's look at how these ratios or functions are evaluated in a right-angled triangle.


Ratios in Trigonometry


Consider a right-angled triangle with the longest side being the hypotenuse and the sides opposite to the hypotenuse being the adjacent and opposite sides.


Trigonometric Functions Formula

If θ is the angle formed by the base and hypotenuse in a right-angled triangle, then

sinθ=PerpendicularHypotenuse

cosθ=BaseHypotenuse

tanθ=PerpendicularBase

The values of the other three functions, cot, sec, and cosec, are determined by the values of tan, cos, and sin.

cotθ=1tanθ=BasePerpendicular

secθ=1cosθ=HypotenuseBase

cosecθ=1sinθ=HypotenusePerpendicular


Even and Odd Trigonometric Functions

Even or odd can be used to describe the trigonometric function.


Odd Trigonometric Functions: If f(-x) = -f(x) and symmetric with respect to the origin, a trigonometric function is said to be odd.


Even Trigonometric Functions: If f(-x) = f(x) and symmetric to the y-axis, a trigonometric function is said to be even.

  • sin(x)=sinx

  • cos(x)=cosx

  • tan(x)=tanx

  • cosec(x)=cosecx

  • sec(x)=secx

  • cot(x)=cotx


Trigonometric Functions in Different Quadrants


Trigonometric Functions in Different Quadrants


I Quadrant

II Quadrant

sinθ increases from 0 to 1

sinθ decreases from 1 to 0

cosθ decreases from 1 to 0

cosθ  decreases from 0 to -1

tanθ increases from 0 to

tanθ increases from  to 0

cotθ decreases from to 0

cotθ decreases from 0 to

secθ increases from 1 to

secθ increases from to -1

cosecθ decreases from to 1

cosecθ decreases from 1 to



III Quadrant

IV Quadrant

sinθ increases from 0 to -1

sinθ increases from -1 to 0

cosθ decreases from -1 to 0

cosθ  increases from 0 to 1

tanθ increases from 0 to

tanθ increases from  to 0

cotθ decreases from to 0

cotθ decreases from 0 to

secθ decreases from -1 to

secθ decreases from to 1

cosecθ decreases from to -1

cosecθ decreases from -1 to


Trigonometric Table - Trigonometry Table Formula

The trigonometric table is made up of interrelated trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent are used to calculate standard angle values. Refer to the below trigonometric table chart to know more about these ratios.


Angles

0

30

45

60

90

sinθ

0

12

12

32

1

cosθ

1

32

12

12

0

tanθ

0

13

1

3

secθ

2

2

23

1

cosecθ

1

23

2

2

cotθ

3

1

13

0

Similarly, we can find the trigonometric ratio values for angles other than 90 degrees, such as 180 degrees, 270 degrees, and 360 degrees.


Important Trigonometric Angles

Trigonometric angles are the angles in a right-angled triangle that can be used to represent various trigonometric functions. 0,30,45,60 and 90 are some of the standard angles used in trigonometry. These angles' trigonometric values can be found directly in a trigonometric table. 180,270, and 360 are some other important angles in trigonometry. The angle of trigonometry can be expressed in terms of trigonometric ratios as follows:

  • θ=sin1(PerpendicularHypotenuse)

  • θ=cos1(BaseHypotenuse)

  • θ=tan1(PerpendicularBase)


Unit Circle


Unit Circle


Because the center of the circle is at the origin and the radius is 1, the concept of unit circle allows us to directly measure the angles of cos, sin, and tan. Assume theta is an angle, and the length of the perpendicular is y and the length of the base is x. The hypotenuse is the same length as the radius of the unit circle, which is 1. As a result, the trigonometry ratios can be written as;

sinθ=y

cosθ=x

tanθ=yx


List of Trigonometric Formulas

There are different formulas in trigonometry depicting the relationships between trigonometric ratios and the angles for different quadrants. The basic trigonometry formulas list is given below:


1. Pythagorean Identities

sin2θ+cos2θ=1

tan2θ+1=sec2θ

cot2θ+1=cosec2θ

sin2θ=2sinθcosθ

cos2θ=cos2θsin2θ

tan2θ=2tanθ1tan2θ

cot2θ=cot2θ12cotθ


2. Sine and Cosine Law in Trigonometry

Sine Law: asinA=bsinB=csinC

Cosine Law: c2=a2+b22abcosC

a2=b2+c22bccosA

b2=a2+c22accosB

The lengths of the triangle's sides are a, b, and c, and the triangle's angle is A, B, and C.


3. Sum and Difference identities

Let u and v be the angles:

sin(u+v)=sin(u)cos(v)+cos(u)sin(v)

cos(u+v)=cos(u)cos(v)sin(u)sin(v)

tan(u+v)=tan(u)+tan(v)1tan(u)tan(v)

sin(uv)=sin(u)cos(v)cos(u)sin(v)

cos(uv)=cos(u)cos(v)+sin(u)sin(v)

tan(uv)=tan(u)tan(v)1+tan(u)tan(v)


4. Trigonometry Identities

sin2θ+cos2θ=1

tan2θ+1=sec2θ

cot2θ+1=cosec2θ


5. Euler's Formula for trigonometry

eix=cosx+isinx

Where x is the angle and i is the imaginary number.

Hence Euler’s formula for sin,cos and tan is:

sinx=eixeix2i

cosx=eix+eix2

tanx=(eixeix)i(eix+eix)


Trigonometry notes on Trigonometry Identities

  • The trigonometry identities are the trigonometry equations that include all of the trigonometry ratios of all the angles

  • Each trigonometric ratio can be expressed in terms of another trigonometric ratio.

  • We can easily find the other value of the trigonometry ratio if we know one of the values of the trigonometry ratio.

  • They can also be used to calculate trigonometric formulas.


Application of Trigonometry

The height of a structure or a mountain is calculated using trigonometry. The height of a building can be easily calculated using trigonometric functions and the distance of a structure from the perspective. It is used in a variety of fields and has no specific applications in solving functional problems. For example, in the development of computer music, trigonometry is used: as you may know, sound travels in waves, and this wave pattern is used in the development of computer music by passing it through a sine or cosine function. 


Trigonometry Examples:

Example 1:  A man stands in front of a 44 foot pole. According to his calculations, the pole cast a shadow that was 13 feet long. Can you assist him in determining the sun's angle of elevation from the shadow's tip?

Ans: Let x be the angle of elevation of the sun,


The Angle of Elevation of the Sun


tanx=4413=3.384

x=tan1(3.384)=1.283

Hence, x in degree is 73.54


Example 2: Find the value of sin75

Ans: Given, sin75

To find the value of sin75 use the formula

sin(A+B)=sinAcosB+cosAsinB

Split 75 such that A=30 and B=45

sin75=sin(30+45)

sin30cos45+cos30sin45

1212+3212

122+322

3+122


Solved problems of Previous Year Question

1. Find the general solution of sinx3sin2x+sin3x=cosx3cos2x+cos3x is _________.

Ans: sinx3sin2x+sin3x=cosx3cos2x+cos3x

2sin2xcosx3sin2x2cos2xcosx+3cos2x=0

sin2x(2cosx3)cos2x(2cosx3)=0

(sin2xcos2x)(2cosx3)=0

sin2x=cos2x

2x=2nπ±(π22x)

x=nπ2+π8


2. Find the value of sin(cot1x)?

Ans: Let cot1x=θ

Hence, x=cotθ

W.K.T 1+cot2θ=cosec2θ

1+x2=cosec2θ

W.K.T cosec θ=1sinθ

1+x2=1sin2θ

sin2θ=11+x2

sinθ=11+x2

sin(cot1x)=11+x2


3. A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 meters and the distance between B and C is 100 meters, then the height of the balloon is given by _________.

Ans: 


The Height of the Balloon


x=hcot3α —(i)

(x+100)=hcot2α —(ii)

x+300)=hcotα —(iii)

From (i) and (ii), we get

100=h(cot3αcot2α)=h(sin2αcos3αcos2αsin3α)sin3αsin2α=hsin(3α2α)sin3αsin2α

On simplifying we get,

100=h(sinαsin3αsin2α) —(iv)

Similarly,

From (ii) and (iii), we get

200=h(cot2αcotα)=hsin(2αα)sin2αsinα

On simplifying we get,

200=h(sinαsin2αsinα) —(v)

Now divide equation (iv) and (v) we get,

sin3αsinα=200100sin3αsinα=2 —(vi)

W.K.T sin3α=3sinα4sin3α

So, From equation (vi) we get,

3sinα4sin3α2sinα=0

4sin3αsinα=0sinα=0 or sin2α=14

sin2α=14=sin2(π6)

α=π6

Hence

h=200sin2α=200sinπ3=20032=1003

So the height of the balloon is 1003


Practice problems

1. Find the value of sec2(tan12)+cosec2(cot13)= _________.

Ans: 15


2. If cos1p+cos1q+cos1r=π then p2+q2+r2+2pqr= ________.

Ans: 1


Conclusion

Although trigonometry does not have many practical applications, it does make it easier to work with triangles. It's an excellent addition to geometry and actual measurements. With trigonometry, you can easily find the height without actually climbing a tree. They have a wide range of applications in real life and are extremely useful to most architects and astronomers. A standard trigonometry table helps solve subject-related problems. The 3 basic measures are sin, cos, and tan, and the remaining three are calculated using the formula given in the above list of formulas.

Study Materials for Trigonometry:

These study materials will aid you in comprehending Trigonometry, ensuring a solid foundation for further mathematical pursuits.


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FAQs on JEE Important Chapter - Trigonometry

1. Who is the founder of trigonometry?

A Greek astronomer, geographer and mathematician, Hipparchus discovered the concept of trigonometry.

2. What Does $\theta$ Mean in Trigonometry?

In trigonometry, $\theta$ is used to represent a measured angle as a variable. It's the angle formed by the horizontal plane and the line of sight from the observer's eye to a higher object. Depending on the object's position, it's called the angle of elevation or the angle of depression. When the object is above the horizontal line, it's called the angle of elevation, and when it's below the horizontal line, it's called the angle of depression.

3. What is the best way to find trigonometric functions?

The ratio of the sides of a right-angled triangle is the trigonometric function. The Pythagorean rule Hypotenuse2 = Altitude2 + Base2 is also applied. In addition, the trigonometric functions have different values for different angles between the hypotenuse and the right triangle's base.