Trigonometry Class 10 Formulas List for NCERT Maths Students

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Below are the provided trigonometry class 10 formulas. Trigonometry involves studying the relationships between angles, lengths, and heights of triangles. It encompasses ratios, functions, identities, and formulas used to solve problems, particularly concerning right-angled triangles.

The applications of trigonometry extend to various fields such as engineering, astronomy, physics, and architectural design. This chapter holds significant importance as it covers various topics, including Linear Algebra, Calculus, and Statistics.

In CBSE Class 10, students are introduced to trigonometry, a completely new and challenging chapter. It requires learning and mastering various formulas to apply them effectively. Below, you can find the tabulated Trigonometry Class 10 formulas.

Trigonometry class 10 formulas

Here are all the essential formulas introduced to Class 10 students. You can refer to these formulas at any time to learn and solve trigonometry-related problems effectively.

The trigonometric formulas for ratios predominantly rely on the three sides of a right-angled triangle, namely the adjacent side or base, perpendicular, and hypotenuse (as shown in the figure above). By applying the Pythagoras theorem to the given right-angled triangle, we get:

(Perpendicular)² + (Base)² = (Hypotenuse)²

⇒ (P)² + (B)² = (H)²

Now, let’s explore the formulas based on trigonometric ratios (sine, cosine, tangent, secant, cosecant, and cotangent).

Learn Trigonometry class 10 formulas

SN.	Variable	Defination
1.	Sin A	Perpendicular/Hypotenuse
2.	Cos A	Base/Hypotenuse
3.	Tan A	Perpendicular/Base
4.	Cot A	Base/Perpendicular
5.	Cosec A	Hypotenuse/Perpendicular
6.	Sec A	Hypotenuse/Base

Relation between different trigonometric Function

SN.	Function	Realtion
1.	tan A	sin A/cos A
2.	Cos A	cos A/sin A
3.	Cosec A	1/sin A
4.	Sec A	1/cos A

Trigonometric Sign Functions

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

Trigonometric Identities

sin²A + cos²A = 1
tan²A + 1 = sec²A
cot²A + 1 = cosec²A

Periodic Identities

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

Complementary Ratios

1. Quadrant I

sin(π/2 − θ) = cos θ
cos(π/2 − θ) = sin θ
tan(π/2 − θ) = cot θ
cot(π/2 − θ) = tan θ
sec(π/2 − θ) = cosec θ
cosec(π/2 − θ) = sec θ

2. Quadrant II

sin(π − θ) = sin θ
cos(π − θ) = -cos θ
tan(π − θ) = -tan θ
cot(π − θ) = – cot θ
sec(π − θ) = -sec θ
cosec(π − θ) = cosec θ

3. Quadrant III

sin(π + θ) = – sin θ
cos(π + θ) = – cos θ
tan(π + θ) = tan θ
cot(π + θ) = cot θ
sec(π + θ) = -sec θ
cosec(π + θ) = -cosec θ

4. Quadrant IV

sin(2π − θ) = – sin θ
cos(2π − θ) = cos θ
tan(2π − θ) = – tan θ
cot(2π − θ) = – cot θ
sec(2π − θ) = sec θ
cosec(2π − θ) = -cosec θ

5. Sum and Difference of Two Angles

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)]

6. Double Angle Formulas

sin 2A = 2 sin A cos A = [2 tan A /(1 + tan²A)]
cos 2A = cos²A – sin²A = 1 – 2 sin²A = 2 cos²A – 1 = [(1 – tan²A)/(1 + tan²A)]
tan 2A = (2 tan A)/(1 – tan²A)

7. Triple Angle Formulas

sin 3A = 3 sinA – 4 sin³A
cos 3A = 4 cos³A – 3 cos A
tan 3A = [3 tan A – tan³A]/[1 − 3 tan²A]

What are Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical functions that “undo” the effect of the regular trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). They are also known as “arc” functions or “antitrigonometric” functions. The inverse trigonometric functions allow us to find the angles associated with specific trigonometric ratios.

The inverse trigonometric functions are denoted by “arc” or “a” followed by the trigonometric function’s abbreviation. For example:

Inverse Sine (arcsin or asin): The inverse of the sine function. It gives us the angle whose sine is a given value. Domain: [-1, 1] Range: [-π/2, π/2]
Inverse Cosine (arccos or acos): The inverse of the cosine function. It gives us the angle whose cosine is a given value. Domain: [-1, 1] Range: [0, π]
Inverse Tangent (arctan or atan): The inverse of the tangent function. It gives us the angle whose tangent is a given value. Domain: (-∞, ∞) Range: (-π/2, π/2)
Inverse Cosecant (arccsc or acsc): The inverse of the cosecant function. It gives us the angle whose cosecant is a given value. Domain: (-∞, -1] ∪ [1, ∞) Range: [-π/2, 0) ∪ (0, π/2]
Inverse Secant (arcsec or asec): The inverse of the secant function. It gives us the angle whose secant is a given value. Domain: (-∞, -1] ∪ [1, ∞) Range: [0, π/2) ∪ (π/2, π]
Inverse Cotangent (arccot or acot): The inverse of the cotangent function. It gives us the angle whose cotangent is a given value. Domain: (-∞, ∞) Range: (0, π)

The inverse trigonometric functions are used to find angles in trigonometric equations and solve various real-world problems involving angles and trigonometric ratios. They are essential tools in calculus, engineering, physics, and other fields where trigonometry is applied.

Tips to Memorize Trigonometry Class 10 Formulas

Understand the Concepts: Before memorizing, make sure you understand the basic concepts and principles behind each formula. This will make it easier to remember and apply them.
Create Acronyms or Mnemonics: Associate each formula with a short acronym or mnemonic that represents the formula’s variables or terms. This can help you recall the formula quickly during exams.
Practice Regularly: Practice solving trigonometry problems using the formulas frequently. Repetition will reinforce your memory.
Use Flashcards: Write the formulas on flashcards and review them regularly. This technique enhances memory retention.
Break Down Formulas: Break complex formulas into smaller parts and focus on memorizing each part separately. Then, combine the parts to remember the entire formula.
Relate Formulas to Real-Life Applications: Connect the trigonometry formulas to real-life scenarios or applications. It will make them more relatable and easier to remember.
Visualize: Create visual aids like diagrams or charts to represent the formulas. Visual cues can enhance memory.
Teach Someone Else: Explaining the formulas to someone else helps reinforce your understanding and memorization.
Use Online Resources: There are many online platforms and mobile apps that offer interactive quizzes and games to help memorize formulas effectively.
Stay Consistent: Dedicate regular study time to review and practice the formulas. Consistency is key to long-term retention.

Remember, practice and understanding are crucial for memorizing trigonometry formulas effectively. With consistent effort, you’ll master these formulas in no time!

Applications of Trigonometry Class 10 Formulas

Trigonometry formulas from Class 10 have various practical applications in everyday life as well as in various fields of science, engineering, and technology. Some of the key applications include:

Architecture and Construction: Trigonometry is used in architectural designs, surveying, and construction to calculate angles, distances, and heights of buildings and structures.
Engineering: Engineers use trigonometry to design and analyze various mechanical, electrical, and civil engineering projects, such as bridges, roads, and buildings.
Astronomy and Navigation: Trigonometry is crucial in astronomy to calculate the positions of celestial objects and navigate using tools like compasses and GPS.
Geography and Cartography: Trigonometry is used in mapmaking, cartography, and land surveying to measure distances and map features accurately.
Computer Graphics and Animation: In computer graphics and animation, trigonometry is essential for calculating angles and positioning objects in 2D and 3D space.
Physics: Trigonometry is used extensively in physics to analyze forces, motion, and wave phenomena.
Music and Sound Engineering: Trigonometry plays a role in analyzing sound waves and designing musical instruments.
Optics: Trigonometry is used in optics to calculate angles of incidence and refraction in lenses and mirrors.
Medicine: Medical imaging techniques like X-rays and CT scans use trigonometry to reconstruct images of the internal body.
Mechanical Engineering: Trigonometry is used to analyze mechanical systems, such as gears, pulleys, and levers.

These are just a few examples of how trigonometry formulas are applied in various fields. Trigonometry is a fundamental branch of mathematics with wide-ranging applications, making it essential knowledge for students in Class 10 and beyond.

Q 1. How to solve trigonometry?

Ans: To solve trigonometry problems, follow these steps:

Understand the problem and identify the given information and what needs to be found.
Label the triangle and use standard labels like “opposite,” “adjacent,” and “hypotenuse” for right-angled triangles.
Choose the appropriate trigonometric ratio (sin, cos, tan) based on what needs to be found.
Apply the trigonometric ratio by substituting known values and setting up the equation.
Use the Pythagoras theorem for right-angled triangles if necessary (a² + b² = c²).
Check your calculator mode (degrees or radians) before using trigonometric functions.
Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angle measures.
Check for extraneous solutions that may not fit the problem’s requirements.
Practice regularly and review trigonometric identities for more complex problem-solving.
Seek help from teachers or online resources if needed. Practice is crucial in mastering trigonometry.

Q 2. How can I learn trigonometry fast?

Ans: To learn trigonometry fast, master basic geometry concepts, understand trigonometric ratios, and memorize key formulas. Regularly practice solving diverse problems, use flashcards, and visualize concepts with diagrams. Watch video tutorials, engage in group study, and teach others to reinforce learning.

Set goals, track progress, and practice under time constraints to improve speed and accuracy. Seek help when needed and stay organized with notes and formulas. Be patient and persistent, as learning trigonometry effectively is a gradual process. With dedication, you can quickly grasp concepts and become proficient in trigonometry problem-solving.

Q 3. What is the formula for a 30 60 90 triangle?

Ans: The formula for the sides of a 30-60-90 triangle is as follows:

Shortest side (opposite 30-degree angle): a
Side opposite the 60-degree angle: b = a√3
Hypotenuse: c = 2a

Remember, this formula is applicable to all 30-60-90 triangles, regardless of their size. If you know the length of one side, you can use these ratios to find the lengths of the other sides. Conversely, if you know the lengths of two sides, you can calculate the third side using these ratios.

Q 4. What angle is depression?

Ans: The term “angle of depression” refers to the angle formed between a horizontal line and the line of sight when an observer looks downward to view an object or point that is at a lower level. In other words, it is the angle measured from the horizontal line to the line of sight directed downward.

The angle of depression is commonly used in various fields, such as surveying, architecture, and physics, to determine the height or distance of an object or point below the observer’s viewpoint. It is the complement of the angle of elevation, which is the angle formed when an observer looks upward to view an object or point that is at a higher level.

Q 5. What is angle of sight?

Ans: The term “angle of sight” is not a standard or widely used trigonometric term. It might be a colloquial or informal expression to refer to either the “angle of elevation” or the “angle of depression,” depending on the context.

1. Angle of Elevation: The angle of elevation is the angle formed between a horizontal line and the line of sight when an observer looks upward to view an object or point that is at a higher level. It is used to determine the height or distance of an object above the observer’s viewpoint.

2. Angle of Depression: The angle of depression, as mentioned earlier, is the angle formed between a horizontal line and the line of sight when an observer looks downward to view an object or point that is at a lower level. It helps determine the height or distance of an object below the observer’s viewpoint.

If you encounter the term “angle of sight” in a specific context, it would be best to clarify with additional context or refer to the specific trigonometric concept it represents, which is either the angle of elevation or the angle of depression.