Trigonometric Formulas Class 11

Trigonometric Formulas are mathematical expressions that relate the angles and sides of triangles. These formulas help in solving problems related to angles, distances, and heights in various geometric and real-world scenarios.

Below is a list of all the trigonometry formulas that will be used in Class 11, according to the NCERT, along with solutions to questions based on them at the end.

Trigonometric Ratios

Trigonometric ratios are the ratios of sides of a right-angle triangle taken two at a time.

A right-angled triangle is a triangle that contains an angle measuring 90°. The side opposite to the right angle is called the hypotenuse, while the sides adjacent to the right angle are called the adjacent and opposite sides.

These formulas define the six trigonometric ratios based on the sides of a right triangle concerning angle θ.

sin (θ) = Perpendicular/Hypotenuse
cos (θ) = Base/Hypotenuse
tan (θ) = Perpendicular/Base = Sin (θ)/cos (θ)
cosec (θ) = Hypotenuse/Perpendicular= 1/sin(θ)
sec (θ) = Hypotenuse/Base = 1/cos (θ)
cot (θ) = Base/Perpendicular =1/tan (θ)

➣ Trick to remember -SOHCAHTOA: [ Students Our Homework Can Help To Overcome Algebra ]

Trigonometry Table

These are the fixed values of trigonometric functions for common angles.

➣ Trick to Remeber Trigonometry table- [ Hand Trick ]

Trigometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all values of the involved variables. These identities are fundamental in simplifying expressions and solving trigonometric equations.

Trigonometry-Identities — List of Trigonometric Identites

Pythagorean Identities

These are derived from the Pythagorean theorem and connect the squares of sine, cosine, and other functions.

sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
cosec² θ = 1 + cot² θ

➣ Read in Detail-Pythagorean Identites

Co-Function Identitites

Co-function identities show how trigonometric functions of negative angles relate to their positive counterparts.

sin (-θ) = -sin θ
cos (-θ) = cos θ
tan (-θ) = -tan θ
cot (-θ) = -cot θ
sec (-θ) = sec θ
cosec (-θ) = -cosec θ

➣ Read in Detail-Co-function Identities

Complementary Angles Identities

Complementary Angle Identities relate the trigonometric functions of an angle to the functions of its complementary angle (i.e., angles that add up to 90° or π/2 radians). These identities are:

sin (90° – θ) = cos θ
cos (90° – θ) = sin θ
tan (90° – θ) = cot θ
cot (90° – θ) = tan θ
sec (90° – θ) = cosec θ
cosec (90° – θ) = sec θ

➣ Read in Detail-Complementary Angle

Supplementary Angles Identities

Two angles are supplementary if their sum is 180° (or π radians). These identities are

sin (180°- θ) = sinθ
cos (180°- θ) = -cos θ
cosec (180°- θ) = cosec θ
sec (180°- θ)= -sec θ
tan (180°- θ) = -tan θ
cot (180°- θ) = -cot θ

➣ Read in Detail-Supplementary Angles

Periodicity of Trigonometric Ratios

These identities show how trig functions repeat after a full cycle of their period:

sin (2nπ + θ) = sin θ
cos (2nπ + θ) = cos θ
tan (nπ + θ) = tan θ
cosec (2nπ + θ) = cosec θ
sec (2nπ + θ) = sec θ
cot (nπ + θ) = cot θ

➣ Read in Detail-Periodicity of Trigonometric Ratios

Angle Sum and Difference Formulas

These formulas allow you to break down or combine trigonometric functions of sums and differences of angles.

A list of angle sum and difference formulas is given as follows:

Sine Formulas:

sin (A+B) = sin A cos B + cos A sin B
sin (A -B) = sin A cos B – cos A sin B

Cosine Formulas:

cos (A + B) = cos A cos B – sin A sin B
cos (A – B) = cos A cos B + sin A sin B

Tangent Formulas:

tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
tan (A-B) = (tan A – tan B)/(1 + tan A tan B)

➣ Read in detail- Sum and Difference Formulas.

Product to Sum Formulas

A list of formulas for converting the product of two trigonometric functions into the sum or difference of two trigonometric functions is given as follows:

sin(A) sin(B) = 1/2 [cos(A−B) − cos(A+B)]
cos(A) cos(B) = 1/2 [cos(A−B) + cos(A+B)]

Some other related identities are:

sin(A+B) sin(A–B) = sin²A – sin²B = cos²B – cos²A
cos(A+B) cos(A–B) = cos²A – sin²B = cos²B – sin²A
sinA + sinB = 2 sin (A+B)/2 cos (A-B)/2

➣ Read in detail-Product to Sum Formulas.

Double Angle Formulas

Identities containing double angles are:

sin (2θ) =2sin(θ)cos(θ) = [2tan θ /(1+tan²θ)]
cos⁡(2θ)=cos⁡²(θ)−sin^⁡2(θ) = 1–2sin²θ = 2cos²θ–1 = [(1-tan²θ)/(1+tan²θ)]
tan⁡(2θ)=2tan⁡(θ) / 1−tan⁡²(θ)

➣ Read in detail- Double angle Formulas

Half Angle Formulas

Half-angle formulas are derived by replacing 2x with x/2 in double-angle identities, resulting in the following identities:

\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}

➣ Read in Detail- Half Angle Formulas

Triple Angle Formulas

Trigonometric Formulas containing three times the multiple of any angle θ are given as:

sin 3θ = 3sinθ – 4sin³θ
cos 3θ = 4cos³θ – 3cosθ
tan 3θ = [3tanθ–tan³θ]/[1−3tan²θ]

➣ Read in Detail-Triple Angle Formulas

Check Other Identities-

Trigonometric Identities	Inverse Trigonometric Identities	Even-Odd formulas

Trigonometric Functions in Different Quadrants

The signs of different functions in the 4 different quadrants, namely, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV, are shown in the table below:

Angle	Quadrant I	Quadrant II	Quadrant III	Quadrant IV
Sin θ	+	+	-	-
cos θ	+	-	-	+
tan θ	+	-	+	-
cot θ	+	-	+	-
sec θ	+	-	-	+
csc θ	+	+	-	-

➣ Trick to remember the Trigonometric Quadants:- " Add Sugar To Coffee"

Where:

Add => All trigonometric functions are positive in the first quadrant.
Sugar => Sine and its reciprocal, cosecant, are positive in the second quadrant.
To => Tan and its reciprocal, cotangent, are positive in the third quadrant.
Coffee => Cosine and its reciprocal, secant, are positive in the fourth quadrant.

Trigonometry Formulas for other Grades:

Solved Examples Using Trigonometric Formulas Class 11

Example 1: Solve for x: sin(2x) = cosx
Solution:

2sinx cosx = cosx
Dividing both sides by cosx (assuming cos⁡x ≠ 0:
2sinx =1
Solving for sinx:
sinx = 1/2
x = π/6 + 2nπ where n is an integer.

Example 2: Simplify the expression: tan²A−sin²A
Solution:

tan²A−sin²A
Using the identity tan²A= sec²A − 1
(sec²A − 1) −sin²A
On simplifying:
sec²A−1−sin²A
sec²A−sin²A−1

Example 3: Express in terms of sine and cosine: tan(x/2)
Solution:

Using the identity tan θ/2 = sinθ / [ 1 + cosθ ]
tan x/2 = sinx / [1 + cosx ]

Example 4: Solve for x: sin(x)⋅cos(x)= 1/2
Solution:

Using the identity sin(2x)=2sin(x)cos(x):
sin(x)⋅cos(x)= 2/1
2sin(x)cos(x)=1
sin(2x)=1
Solving for x:
2x= 2/π+2nπ
x= 4/π+nπ where n is an integer.

Example 5: Simplify the expressionsec²(x) - tan²(x)

Solution:

Using the identity sec²(x)=1+tan²(x):
sec²(x)−tan²(x)=(1+tan²(x))−tan²(x)
sec²(x)−tan²(x)=1

Practice Question on Trigonometric Formulas Class 11

Question: 1 Simplify sin⁡(90°− θ) + cos⁡(90° − θ).

Question: 2 Prove the identity (1 − cos⁡2A)/sin⁡2A = tan⁡A

Question: 3 Evaluate using identity cos⁡(180°− θ) + cos⁡(180°+ θ)

Question: 4 If tan⁡A = 3/4, find the values of:

sin⁡2A
cos⁡2A

Question: 5 Using co-function identity, prove that: csc⁡(90°−θ) = sec⁡θ

Question: 6 Find the value: tan⁡(45° + 30°)

Question: 7 Prove the identity: \frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta} = \sin^2 \theta + \cos^2 \theta

Question: 8 If sin(A + B) = sinA cosB + cosA sinB, prove that: cos(A - B) = cosA cosB + sinA sinB.

Answer Key:
cosθ + sinθ
Proven using double angle identities
−2cosθ
3/4
csc⁡(90°−θ) = sec⁡θ (True using co-function identity)
(√6 + √2)/4