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Jonathan Kelly
Jonathan Kelly

Trigonometry Formulas for Class 11 and 12: A Complete List with PDF Download


` tags for the title of your article. - Use `` tags for the main headings of each section. Trigonometry Formulas for Class 11 and 12: A Complete Guide




Do you want to learn how to master trigonometry formulas for class 11 and 12? If yes, then you have come to the right place. In this article, I will explain everything you need to know about trigonometry formulas for class 11 and 12, including their definitions, derivations, applications, examples, and proofs. By the end of this article, you will be able to use trigonometry formulas for class 11 and 12 with confidence and ease. But first, let's start with some basics. What is trigonometry and why is it important for students of class 11 and 12? Basic Trigonometric Functions




Trigonometry is a branch of mathematics that deals with the relationships between angles and lengths of triangles. It is derived from the Greek words "trigonon" (triangle) and "metron" (measure). Trigonometry is useful for many applications, such as astronomy, navigation, engineering, physics, and geometry. The six basic trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. They are defined as follows: - Sine (sin): The ratio of the opposite side to the hypotenuse of a right-angled triangle. - Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right-angled triangle. - Tangent (tan): The ratio of the opposite side to the adjacent side of a right-angled triangle. - Secant (sec): The reciprocal of cosine, or the ratio of the hypotenuse to the adjacent side of a right-angled triangle. - Cosecant (csc): The reciprocal of sine, or the ratio of the hypotenuse to the opposite side of a right-angled triangle. - Cotangent (cot): The reciprocal of tangent, or the ratio of the adjacent side to the opposite side of a right-angled triangle. The following diagram shows how these functions are related to a right-angled triangle with an angle θ:


We can use the Pythagorean theorem to find the values of these functions for any angle θ in a right-angled triangle. The Pythagorean theorem states that: a + b = c where a and b are the lengths of the legs and c is the length of the hypotenuse. Using this theorem, we can derive some useful relationships between the trigonometric functions: sinθ + cosθ = 1 tanθ + 1 = secθ cotθ + 1 = cscθ Here are some examples of how to use these functions to solve problems involving angles and lengths of triangles: Example 1: Find the value of sin 30. Solution: We can use a special right-angled triangle with angles 30, 60, and 90 to find the value of sin 30. This triangle has sides in the ratio 1:3:2. Therefore, sin 30 = opposite/hypotenuse = 1/2 Example 2: Find the value of cos π/4. Solution: We can use another special right-angled triangle with angles π/4, π/4, and π/2 to find the value of cos π/4. This triangle has sides in the ratio 1:1:2. Therefore, cos π/4 = adjacent/hypotenuse = 1/2 Example 3: Find the value of tan 45. Solution: We can use the same triangle as in example 2 to find the value of tan 45. Therefore, tan 45 = opposite/adjacent = 1/1 = 1 Example 4: Find the length of x in the following triangle:


Solution: We can use any trigonometric function that involves x and one of the given angles or sides. For example, we can use cosine: cos 53 = adjacent/hypotenuse = x/13 x = 13 cos 53 Trigonometric Identities




A trigonometric identity is an equation that is true for all values of the variable or angle involved. Trigonometric identities are useful for simplifying expressions and solving equations involving trigonometric functions. There are many trigonometric identities, but some of the most common ones are: - Reciprocal identities: These identities relate the reciprocal functions (sec, csc, and cot) to the basic functions (cos, sin, and tan). sec θ = 1/cos θ csc θ = 1/sin θ cot θ = 1/tan θ - Quotient identities: These identities relate the quotient functions (tan and cot) to the basic functions (cos and sin). tan θ = sin θ/cos θ cot θ = cos θ/sin θ - Pythagorean identities: These identities are derived from the Pythagorean theorem and relate the squares of the basic functions. sinθ + cosθ = 1 tanθ + 1 = secθ cotθ + 1 = cscθ - Co-function identities: These identities relate the functions of complementary angles (angles that add up to 90 or π/2). sin(90 - θ) = cos θ cos(90 - θ) = sin θ tan(90 - θ) = cot θ sec(90 - θ) = csc θ csc(90 - θ) = sec θ cot(90 - θ) = tan θ - Periodic identities: These identities relate the functions of angles that differ by a multiple of 180 or π. sin(θ + π) = -sin θ cos(θ + π) = -cos θ tan(θ + π) = tan θ sec(θ + π) = -sec θ csc(θ + π) = -csc θ cot(θ + π) = cot θ Here are some examples of how to use these identities to prove or verify other identities or equations: Example 5: Prove that sin(π/4) + cos(π/4) = 1. Solution: We can use the Pythagorean identity to prove this: sin(π/4) + cos(π/4) = (sin π/4) + (cos π/4) = (1/2) + (1/2) = 1/2 + 1/2 = 1 Example 6: Verify that tan(60) = 3. Solution: We can use the quotient identity and the co-function identity to verify this: tan(60) = sin(60)/cos(60) = sin(90 - 30)/cos(90 - 30) = cos(30)/sin(30) = (3/2)/(1/2) Trigonometric Formulas




A trigonometric formula is an equation that relates two or more trigonometric functions or angles. Trigonometric formulas are useful for finding the values of trigonometric functions or angles in different situations, such as solving triangles, graphing functions, or modeling periodic phenomena. There are many trigonometric formulas, but some of the most important ones are: - Sum and difference formulas: These formulas relate the functions of the sum or difference of two angles to the functions of the individual angles. sin(α + β) = sin α cos β + cos α sin β sin(α - β) = sin α cos β - cos α sin β cos(α + β) = cos α cos β - sin α sin β cos(α - β) = cos α cos β + sin α sin β tan(α + β) = (tan α + tan β)/(1 - tan α tan β) tan(α - β) = (tan α - tan β)/(1 + tan α tan β) - Double angle formulas: These formulas relate the functions of twice an angle to the functions of the angle. sin 2θ = 2 sin θ cos θ cos 2θ = cosθ - sinθ = 2 cosθ - 1 = 1 - 2 sinθ tan 2θ = (2 tan θ)/(1 - tanθ) - Half angle formulas: These formulas relate the functions of half an angle to the functions of the angle. sin θ/2 = [(1 - cos θ)/2] cos θ/2 = [(1 + cos θ)/2] tan θ/2 = [(1 - cos θ)/(1 + cos θ)] = (sin θ)/(1 + cos θ) = (1 - cos θ)/(sin θ) - Product to sum formulas: These formulas relate the product of two functions to the sum or difference of two functions. sin α sin β = (1/2)[cos(α - β) - cos(α + β)] cos α cos β = (1/2)[cos(α - β) + cos(α + β)] sin α cos β = (1/2)[sin(α + β) + sin(α - β)] cos α sin β = (1/2)[sin(α + β) - sin(α - β)] - Sum to product formulas: These formulas relate the sum or difference of two functions to the product of two functions. sin α sin β = 2 sin[(α β)/2] cos[(α β)/2] cos α + cos β = 2 cos[(α + β)/2] cos[(α - β)/2] cos α - cos β = -2 sin[(α + β)/2] sin[(α - β)/2] Here are some examples of how to use these formulas to find the values of trigonometric functions or angles in different situations: Example 7: Find the value of sin 75. Solution: We can use the sum formula for sine to find the value of sin 75: sin 75 = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30 = (1/2)(3/2) + (1/2)(1/2) = (3 + 1)/(22) Example 8: Find the value of cos π/8. Solution: We can use the half angle formula for cosine to find the value of cos π/8: cos π/8 = [(1 + cos π/4)/2] = [(1 + 1/2)/2] = [(2 + 1)/(22)] Example 9: Find the value of tan(15 + x), given that tan x = 3. Solution: We can use the sum formula for tangent to find the value of tan(15 + x): tan(15 + x) = (tan 15 + tan x)/(1 - tan 15 tan x) Inverse Trigonometric Functions




An inverse trigonometric function is a function that reverses the process of finding an angle from a trigonometric function value. For example, if we know the value of sin θ, we can use the inverse sine function, denoted by sin or arcsin, to find the value of θ. Similarly, we can use the inverse cosine, inverse tangent, inverse secant, inverse cosecant, and inverse cotangent functions to find the angles corresponding to the values of cos, tan, sec, csc, and cot respectively. The inverse trigonometric functions have certain domains and ranges that restrict the possible values of the angles. For example, the domain of sin is [-1, 1], and the range is [-π/2, π/2]. This means that we can only find the angles between -90 and 90 using sin. The following table summarizes the domains and ranges of the six inverse trigonometric functions: Function Domain Range --- --- --- sin [-1, 1] [-π/2, π/2] cos [-1, 1] [0, π] tan (-, ) (-π/2, π/2) sec (-, -1] [1, ) [0, π/2) (π/2, π] csc (-, -1] [1, ) [-π/2, 0) (0, π/2] cot (-, ) (0, π) We can find the values of inverse trigonometric functions using tables or calculators. However, we should be careful about the units and modes of the angles. For example, some calculators use radians instead of degrees, or principal angles instead of general angles. We should also be aware of the possible ambiguities or multiple solutions that may arise from using inverse trigonometric functions. For example, if we use sin(0.5), we may get 30 or 150 as possible answers. Here are some examples of how to use inverse trigonometric functions to solve problems involving angles or triangles: Example 10: Find the value of x in the following equation: sin x = 0.8 Solution: We can use sin to find the value of x: x = sin(0.8) x 53.13 However, this is not the only possible solution. We can also find another angle in the range [-π/2, π/2] that has the same sine value by using the co-function identity: x = 90 - sin(0.8) x 36.87 Example 11: Find the value of y in the


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