What is the importance of trigonometry maths for JEE?

Trigonometry is crucial when solving wave, oscillatory, and circular motion problems in physics. It appears in applications of calculus, coordinate geometry, and further mathematics as well.

What are the basic trigonometric ratios?

The basic trigonometric ratios are sine, cosine, tangent, cotangent, secant, and cosecant, which are all derived from a right-angled triangle.

What are trigonometric identities, and why are they important?

Trigonometric identities are equations that link trigonometric functions together. They reduce complex trigonometric expressions and form the backbone of solving any problem in JEE.

What is the unit circle, and why is it important in trigonometry?

A unit circle is a circle with radius 1, and it defines the trigonometric functions for all angles. Beyond 90°, this is critical for understanding what happens to the functions.

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Trigonometry

Trigonometry is that branch of mathematics that relates to the sides and angles of triangles, especially in a right-angled triangle. In solving a vast array of problems, one has to learn the basic principles of trigonometric functions and identities. Below, we discuss the major topics in trigonometry IIT JEE maths and how they help in JEE preparation.

1.0Key Concepts of Trigonometry

Trigonometric Ratios and identities

In a right-angled triangle, there exist six trigonometric ratios and identities fundamental defining the relationship between the angles and the sides:

$Sine (sin) : sin θ = \frac{Perpendicular}{Hypotenuse}$

$Cosine (cos) : cos θ = \frac{Base}{Hypoternuse}$

$Tangent (tan) : tan θ = \frac{Perpendicular}{Base}$

$Cotangent (cot) : cot θ = \frac{1}{t a n θ} = \frac{Base}{Perpendicular}$

$Secant (sec): sec θ = \frac{1}{c o s θ} = \frac{Hypotenuse}{Base}$

$Cosecant (csc): cosec θ = \frac{1}{s i n θ} = \frac{Hypotenuse}{Perpendicular}$

	0	30	45	60	90
Sine	0	1 / 2	$1/ 2$	$3 /2$	1
cos	1	$3 /2$	$1/ 2$	1 / 2	0
tan	0	$1/ 3$	1	$3$	ND
sec	1	$2/ 3$	$2$	2	ND
cosec	ND	2	$2$	$2/ 3$	1
cot	ND	$3$	1	$1/ 3$	0

Also Read Trigonometry Previous Year Questions with Solutions

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying trigonometric expressions and solving equations. Some important identities include:

Pythagorean Identity

$sin^{2} θ + cos^{2} θ = 1$

$1 + tan^{2} θ = sec^{2} θ$

$1 + cot^{2} θ = cosec^{2} θ$

Reciprocal Identities

$sec θ = \frac{1}{c o s θ} \cdot cosec θ = \frac{1}{s i n θ}, cot θ = \frac{1}{t a n θ}$

Sum and Difference Identities

$sin (A \pm B) = sin A cos B \pm cos A sin B$

$cos (A \pm B) = cos A cos B \mp sin A sin B$

$tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$

$cot (A \pm B) = \frac{c o t A \cdot c o t B \mp 1}{c o t B \pm c o t A}$

$sin (A + B) sin (A - B) = sin^{2} A - sin^{2} B = cos^{2} B - cos^{2} A$

$cos (A + B) cos (A - B) = cos^{2} A - sin^{2} B = cos^{2} B - sin^{2} A$

\sin (A+B+C)=\sinA \cos B \cos C+\sin B \cos A \cos C+\sin C \cos A \cos B-\sin A \sin B \sin C

$cos (A + B + C) = cos A cos B cos C - cos A sin B sin C - cos B sin A sin C - cos C sin A sin B$

$tan (A + B + C) = \frac{t a n A + t a n B + t a n C - t a n A t a n B t a n C}{1 - t a n A t a n B - t a n B t a n C - t a n C t a n A}$

Transformation Formula

$2 sin A cos B = sin (A + B) + sin (A - B)$

$2 sin B cos A = sin (A + B) - sin (A - B)$

$2 cos A cos B = cos (A + B) + cos (A - B)$

$2 sin A sin B = cos (A - B) - cos (A + B)$

$sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$

$sin A - sin B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$

$cos A + cos B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$

$cos A - cos B = 2 sin (\frac{A + B}{2}) sin (\frac{A - B}{2})$

Multiple Angle Ratios

$sin 2 A = 2 sin A cos A = \frac{2 t a n A}{1 + t a n ^{2} A}$

$cos 2 A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A = \frac{1 - t a n ^{2} A}{1 + t a n ^{2} A} = cos^{2} A - sin^{2} A$

$tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}$

$sin 3 A = 3 sin A - 4 sin^{3} A$

$cos 3 A = 4 cos^{3} A - 3 cos A$

$tan 3 A = \frac{3 t a n A - t a n ^{3} A}{1 - 3 t a n ^{2} A}$

Also Read Inverse Trigonometric Functions

Trigonometric Functions of Angles

The trigonometric functions are further defined on the unit circle. This defines the concept of sine, cosine, tangent, and other related functions for all angles and just right triangles. The periodicity of the trigonometric circular function is deeper with a unit circle.

Unit Circle Definition: For an angle, , the sine and cosine are defined as:

$sin θ = y -coordinate, cos θ = x - coordinate,$

and

$tan θ = \frac{s i n θ}{c o s θ}$

This can be visualized by seeing the behaviour of the values of the trigonometric functions for angles larger than 90°, that is, in the second, third, and fourth quadrants.

Also Read Integration Trigonometric Functions

Trigonometric Equations

The Trigonometric equations often involve using the basic trigonometric ratios and identities for determining the measures of unknown angles. For instance, general equations for these trigonometric functions are:

Sinθ = Sinα, and the general solution is θ = nπ + (-1)nα, where n ∈ Z

Cosθ = Cosα, and the general solution is θ = 2nπ + α, where n ∈ Z

Tanθ = Tanα, and the general solution is θ = nπ + α, where n ∈ Z

2.0Graphs of Trigonometric Functions

Trigonometric Function	Graphs
Sine y=sinx Domain is R Range (-1,1)
Cos y=cosx Domain is R Range (-1,1)
Tan y=tanx Domain = R - (2n+1)π/2 Range = R
Cot y= cotx Domain = R - nπ Range = R
Sec y= secx Domain: $R - (2 x + 1) π /2$ Range $(- \infty, - 1) \cup (1, \infty)$
Cosec y = cosecx Domain: $R - nπ$ Range : $(- \infty, - 1) \cup (1, \infty)$

Trigonometric Function

Graphs

Sine

y=sinx

Domain is R

Range (-1,1)

Cos

y=cosx

Domain is R

Range (-1,1)

Tan

y=tanx

Domain = R - (2n+1)π/2

Range = R

Cot

y= cotx

Domain = R - nπ

Range = R

Sec

y= secx

Domain: $R - (2 x + 1) π /2$

Range $(- \infty, - 1) \cup (1, \infty)$

Cosec

y = cosecx

Domain: $R - nπ$

Range : $(- \infty, - 1) \cup (1, \infty)$

3.0Solved Problems

Question 1: If

$Sin x = - \frac{3}{5}, where π < x < \frac{3 π}{2}, then 80 (tan^{2} x - cos x)$ is equal to

Ans: 109

Explanation

$sin x = - \frac{3}{5} where π < x < \frac{3 π}{2}$

$tan x = \frac{3}{4}, cos x = \frac{- 4}{5}$

$80 (tan^{2} x - cos x)$

$= 80 (\frac{9}{16} + \frac{4}{5})$

$= (\frac{45 + 64}{80})$

=109

Question2: Suppose

$θ \in [0, \frac{π}{4}] is a solution of 4 cos θ - 3 sin θ = 1 .$

Then

$cos θ$ is equal to:

Ans:

$\frac{4}{( 3 6 - 2 )}$

Explanation

$4 cos θ - 3 sin θ = 1$

$4 cos θ - 1 = 3 sin θ$

Squaring Both sides

$16 cos^{2} θ - 8 cos θ + 1 = 9 (1 - cos^{2} θ)$

$16 cos^{2} θ - 8 cos θ - 8 = - 9 cos^{2} θ$

$\Rightarrow 25 cos^{2} θ - 8 cos θ - 8 = 0$

$\Rightarrow cos θ = \frac{8 \pm 64 + 4 \times 25 \times 8}{2.25}$

$= \frac{8 \pm 4 4 + 50}{2.25}$

$= \frac{4 \pm 2 54}{25}$

$As θ \in [0, \frac{π}{4}]$

$\Rightarrow cos θ = \frac{4 + 6 6}{25} = \frac{4}{3 6 - 2}$

Question 3: Show that 2 sin²β + 4 cos (α + β) sin α sin β + cos 2 (α + β) = cos 2α

Solution: LHS

= 2 sin²β + 4 cos (α + β) sin α sin β + cos 2(α + β)

= 2 sin²β + 4 (cos α cos β – sin α sin β) sin α sin β + (cos 2α cos 2β – sin 2α sin 2β)

= 2 sin²β + 4 sin α cos α sin β cos β – 4 sin²α sin²β + cos 2α cos 2β – sin 2α sin 2β

(4 sin α cos α sin β cos β = 2 sin α cos α .2sin β cos β)

(2 sin α cos α =sin 2α and 2sin β cos β=sin 2β)

= 2 sin²β + sin 2α sin 2β – 4 sin²α sin²β + cos 2α cos 2β – sin 2α sin 2β

(cos 2β = 1–2 sin²β)

= (1 – cos 2β) – (2 sin²α) (2 sin²β) + cos 2α cos 2β

= (1 – cos 2β) – (1 – cos 2α) (1 – cos 2β) + cos 2α cos 2β

= (1 – cos 2β)(1–1+cos 2α) + cos 2α cos 2β

= (1 – cos 2β)(cos 2α) + cos 2α cos 2β

= cos 2α – cos 2α cos 2β + cos 2α cos 2β

= cos 2α

Problem 4: Prove that Sec8-1sec4-1=tan8tan2

Solution: LHS

Sec8-1sec4-1=1/cos8-11/cos4-1

=(1-cos8)cos4cos8(1-cos4)

(cos2 = 1 – sin²; sin² = 1 – cos2; sin²4 = 1 – cos8)

=2sin24cos4cos8sin22

=sin4(2sin4cos4)2cos8sin22

(sin2 = 2sincos; sin8 = 2sin4cos4)

=sin4sin82cos8sin22

=2sin2cos2sin8sin22cos8

(sincos=tan)

= tan8tan2

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Trigonometry

Trigonometry is that branch of mathematics that relates to the sides and angles of triangles, especially in a right-angled triangle. In solving a vast array of problems, one has to learn the basic principles of trigonometric functions and identities. Below, we discuss the major topics in trigonometry IIT JEE maths and how they help in JEE preparation.