Trigonometry Formulas

What is Trigonometry Formulas?

Trigonometry formulas are mathematical equations that relate the angles and sides of triangles. These formulas are essential for solving problems in various fields, including physics, engineering, and architecture. Key trigonometric formulas include the sine, cosine, and tangent functions.

Basic Trigonometry Formulas

Basic trigonometry formulas establish the relationship between trigonometric ratios and the corresponding sides of a right-angled triangle. There are six primary trigonometric ratios, also known as trigonometric functions: sine, cosine, secant, cosecant, tangent, and cotangent. These functions are abbreviated as sin, cos, sec, csc, tan, and cot, respectively. These functions and identities are derived using a right-angled triangle as the reference.

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Trigonometric Ratio Formulas

• sin θ = Perpendicular/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Perpendicular/Base
• sec θ = Hypotenuse/Base
• cosec θ = Hypotenuse/Perpendicular
• cot θ = Base/Perpendicular

Reciprocal Identities

Cosecant, secant, and cotangent are the reciprocals of the basic trigonometric ratios: sine, cosine, and tangent, respectively. These reciprocal identities are derived from the properties of a right-angled triangle and play a crucial role in trigonometry. They are often used to simplify and solve trigonometric problems. The formulas for these reciprocal trigonometric identities, which are essential for various calculations and transformations in trigonometry, are:

• cosec θ = 1/sin θ; sin θ = 1/cosec θ
• sec θ = 1/cos θ; cos θ = 1/sec θ
• cot θ = 1/tan θ; tan θ = 1/cot θ

Pythagorean Identities

The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse (ccc) is equal to the sum of the squares of the other two sides (aaa and bbb). Mathematically, this is expressed as: c^² = a² + b^² ,Using this theorem, we can derive Pythagorean identities in trigonometry, which allow us to convert one trigonometric ratio into another. These identities are fundamental in simplifying and solving trigonometric equations.

• sin²θ + cos^²θ = 1
• sec²θ – tan²θ = 1
• csc²θ – cot²θ = 1

Trigonometric Ratio Table

Periodicity Identities (in Radians):

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A &  cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

Cofunction Identities (in Degrees):

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

Sum & Difference Identities

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• 𝑡𝑎𝑛(𝑥+𝑦)= (𝑡𝑎𝑛 𝑥+𝑡𝑎𝑛𝑦) / (1−𝑡𝑎𝑛 𝑥.𝑡𝑎𝑛 𝑦)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x-y) = (tan x – tan y) (1+tan x.tan y)

Triple Angle Identities

• Sin 3x = 3sin x – 4sin³x
• Cos 3x = 4cos³x-3cos x
• Tan 3x = (3tanx – tan³x)/(1-3tan²x)

Inverse Trigonometry Formulas

• sin⁻¹ (–x) = – sin⁻¹ x
• cos⁻¹ (–x) = π – cos⁻¹ x
• tan⁻¹ (–x) = – tan⁻¹ x
• cosec⁻¹ (–x) = – cosec^⁻¹ x
• sec⁻¹ (–x) = π – sec⁻¹ x
• cot^⁻¹ (–x) = π – cot⁻¹ x

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

Trigonometry Formulas For Class 10

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

1. sin²A + cot²A = 1
2. tan²A + 1 = sec²A
3. 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

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

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

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

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

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)]

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)

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]

Trigonometry Formulas For Class 11

Trigonometry Formulas

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

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]
cos x cos y = 1/2[cos(x–y) + cos(x+y)]
sin x cos y = 1/2[sin(x+y) + sin(x−y)]
cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]
sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]
cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]
cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Identities

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

Quadrants→	I	II	III	IV
Sin A	+	+	–	–
Cos A	+	–	–	+
Tan A	+	–	+	–
Cot A	+	–	+	–
Sec A	+	–	–	+
Cosec A	+	+	–	–

Trigonometry Formulas For Class 12

Domain and Range of Trigonometric Functions

Here are the domain and range of basic trigonometric functions:

• Sine function, sine: R → [– 1, 1]
• Cosine function, cos : R → [– 1, 1]
• Tangent function, tan : R – { x : x = (2n + 1) π/2, n ∈ Z} →R
• Cotangent function, cot : R – { x : x = nπ, n ∈ Z} →R
• Secant function, sec : R – { x : x = (2n + 1) π/2, n ∈ Z} →R – (– 1, 1)
• Cosecant function, cosec : R – { x : x = nπ, n ∈ Z} →R – (– 1, 1)

Properties of Inverse Trigonometric Functions

• sin⁻¹ (1/a) = cosec⁻¹(a), a ≥ 1 or a ≤ – 1
• cos⁻¹(1/a) = sec^⁻¹(a), a ≥ 1 or a ≤ – 1
• tan⁻¹(1/a) = cot^⁻¹(a), a>0
• sin⁻¹(–a) = – sin^⁻¹(a), a ∈ [– 1, 1]
• tan⁻¹(–a) = – tan^⁻¹(a), a ∈ R
• cosec^⁻¹(–a) = –cosec⁻¹(a), | a | ≥ 1
• cos^⁻¹(–a) = π – cos^⁻¹(a), a ∈ [– 1, 1]
• sec^⁻¹(–a) = π – sec^⁻¹(a), | a | ≥ 1
• cot^⁻¹(–a) = π – cot^⁻¹(a), a ∈ R

Addition Properties of Inverse Trigonometry functions

• sin^⁻¹a + cos^⁻¹a = π/2, a ∈ [– 1, 1]
• tan^⁻¹a + cot^⁻¹a = π/2, a ∈ R
• cosec^⁻¹a + sec^⁻¹a = π/2, | a | ≥ 1
• tan^⁻¹a + tan^⁻¹ b = tan^⁻¹ [(a+b)/1-ab], ab<1
• tan^⁻¹a – tan^⁻¹ b = tan^⁻¹ [(a-b)/1+ab], ab>-1
• tan^⁻¹a – tan^⁻¹ b = π + tan^⁻¹[(a+b)/1-ab], ab > 1; a,b > 0

Twice of Inverse of Tan Function

• 2tan^⁻¹a = sin^⁻¹ [2a/(1+a²)], |a| ≤ 1
• 2tan^⁻¹a = cos^⁻¹[(1-a²)/(1+a²)], a ≥ 0
• 2tan^⁻¹a = tan⁻¹[2a/(1+a²)], – 1 < a < 1

Practice paper

Problem 1: Solving for an Unknown Side

Problem: In a right-angled triangle, if one angle is 30⁰ and the hypotenuse is 10 units, find the length of the side opposite the 30⁰ angle.

Solution:

1. Identify the formula: sin⁡θ = opposite​/hypotenuse
   Plug in the values: sin30⁰ = opposite​/10
   Solve for the opposite side: sin30⁰=1/2
   =>1/2=opposite/10
   Opposite = 10 x 1/2 = 5 units

Problem 2: Finding an Angle Using Cosine

Problem: In a right-angled triangle, if the adjacent side to the angle θ is 8 units and the hypotenuse is 10 units, find θ

Solution:

1. Identify the formula: cosθ = adjacent​/hypotenuse
2. Plug in the values: cosθ = 8/10
3. Simplify the fraction: cosθ=4/5 = 0.8
4. Use the inverse cosine function to find θ:
   θ = cos⁻¹(0.8)≈36.87⁰

Problem 4: Using the Pythagorean Identity

Problem: Given that sin⁡θ = 3/5, find cos⁡θ using the Pythagorean identity.

Solution:

Identify the Pythagorean identity: sin²θ+cos²θ=1

Plug in the values: (3/2)² + cos²θ = 1

Solve for cos⁡²θ: (3/2)² = 9/5

9/5+cos²θ = 1

cos²θ = 1− 9/5 = 16/25

cosθ = ± √(16/25) = ±(4/5)

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