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. In this article we will learn about the different trigonometric formulas of class 11 level and at the last of this we will also solve some example for better understanding.

What are Trigonometric Ratios?

Trigonometric ratios are the ratio of sides of a right-angle triangle taken two at a time. If we take the perpendicular and hypotenuse, we get the sine ratio, which is defined as the ratio of the length of the side opposite the angle (perpendicular) to the length of the hypotenuse of the right triangle. Similarly,

• 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)

Trigonometric Formula Class 11

List of some of the basic trigonometric formulas include the following formulas:

Pythagorean Identitis

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

Co-Function Identitites

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

Supplementary Angles Identities

• 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

Complementary Angles Identities

• 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

Periodicity of Trigonometric Ratios

• 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

Angle Sum and Difference Formulas

List of angle sum and difference formulas is give as follows:

• sin (A+B) = sin A cos B + cos A sin B
• sin (A -B) = sin A cos B – cos A sin B
• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• 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 more about Sum and Difference Formulas.

Product to Sum Formulas

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

Twice Angle Formulas

Identities containing double angle 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 More about Sin 2x Formula.

Half Angle Formulas

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

• [Tex]\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos(\theta)}{2}}[/Tex]
• [Tex]\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}[/Tex]
• [Tex]\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos(\theta)}{1 + \cos(\theta)}}[/Tex]

Thrice Angle Formulas

Trigonometric Formulas containing three time multiple of any angle θ is given as:

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

Sign of Trigonometric Functions in Different Quadrants

The sign 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	+	+	–	–

Applications of Trigonometric Formulas

Some of the key applicaitons of these discussd formulas for class 11 trigonometry are:

• Engineering Design: Trigonometric ratios aid in designing structures, calculating angles, and determining load distributions.

• Physics and Mechanics: Trigonometry is crucial in analyzing forces, motion, and vectors in physics and mechanical systems.

• Navigation and Geography: Trigonometric functions are used in navigation for determining distances, angles, and locations on maps and charts.

• Electrical Engineering: Trigonometric formulas help in analyzing alternating current (AC) circuits and understanding phase differences.

• Computer Graphics: Trigonometry is fundamental in creating realistic computer graphics, determining angles and positions of objects.

Read More,

• Trigonometric Identities
• Inverse Trigonometric Identities 
• Trigonometric Identities Class 10

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 = 6π​+nπ 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 ]

Practice Questions For Class 11 Trigonometry Formulas

Q1: Simplify: sin²x + cos²x

Q2: Evaluate: tan 45° · cot 45°

Q3: If sin A = 3/5 and cos B = 4/5, find tan(A + B).

Q4: Express sin 2x in terms of tan x and cos x

Q5: If cos(θ) = -(1/2) and (θ) is in the second quadrant, find sin(θ)

Trigonometry Formula Class 11: FAQs

What is a Trigonometric Function?

A trigonometric function is a mathematical function that relates the angles of a right triangle to the ratios of its sides, such as sine, cosine, and tangent.

Is there Trigonometry in Class 11?

Yes, trigonometry is typically included in the mathematics curriculum for Class 11 in many educational systems, including CBSE (Central Board of Secondary Education) in India.

What are the Topics of Trigonometry in Class 11?

Topics in trigonometry covered at this level often include trigonometric functions, identities, equations, and their applications.

What are the Important Trigonometry Formulas Class 11?

Some important trigonometry formulas for Class 11 include:

• sin(A±B) = sinAcosB ± cosAsinB
• cos(A±B) = cosAcosB ∓ sinAsinB
• tan(A±B) = tanA ± tanB / 1 ∓ tanA tanB​
• sin(2A) = 2sinA cosA
• ccos(2A) = cos²A−sin²A
• tan(2A) = 2tanA / [ 1−tan²A​ ]

How to memorize Trigonomeric Formulas Class 11?

The best way to memorize class 11 formula is to read out the formulas mentioned above daily and solve practice questions based on them