# Trigonometry Formulas

Created by: Team Maths - Examples.com, Last Updated: May 21, 2024

## 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.

### 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: c² = a² + b² ,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 + cos²θ = 1
• sec²θ – tan²θ = 1
• csc²θ – cot²θ = 1

## Trigonometric Ratio Table

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

• 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

• 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

• 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

• 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

## 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)

## How to Do Trigonometry Easily?

Break down problems into steps, memorize basic formulas, and use mnemonic devices like SOH CAH TOA. Practice regularly with simple problems before advancing to complex ones. Visual aids and triangle diagrams can also help in understanding trigonometric concepts.

## How to Calculate Trigonometry?

Use trigonometric ratios: sine, cosine, and tangent. For any right-angled triangle, apply SOH CAH TOA to find unknown sides or angles. For non-right triangles, use the sine rule or cosine rule. Practice with problems to improve calculation skills.

## What Does SOH CAH TOA Mean?

SOH CAH TOA is a mnemonic to remember trigonometric ratios:

Sine = Opposite / Hypotenuse

## How to Solve Trigonometry Without a Calculator?

Use trigonometric tables or known values for specific angles (like 30°, 45°, 60°). For exact values, remember special triangles and key trigonometric identities. Practice mental math and estimation for other angles.

## What Is the Formula to Memorize Trigonometry?

Use SOH CAH TOA for right triangles. For non-right triangles, remember the sine rule
(a/sinA​=b/sinB​=c/sinC)and cosine rule (c² = a² + b² – 2ab cosC)

## How to Solve Tangent?

To solve for tangent (tanθ)
Identify the opposite and adjacent sides.
Use the formula tan θ = Opposite/adjacent

Plug in the values and simplify.

## How to Find Hypotenuse?

In a right-angled triangle, use the Pythagorean theorem:
Or, use trigonometric ratios if an angle and a side are known.

## How Do 30-60-90 Triangles Work?

In a 30-60-90 triangle, the sides have a consistent ratio:

Shortest side (opposite 30°) = x
Hypotenuse (opposite 90°) = 2x
Longest side (opposite 60°) = x √3

## What Is the Angle of Depression?

The angle of depression is the angle between the horizontal line and the line of sight looking downward. It is equal to the angle of elevation from the observer’s point of view on the ground.

## How to Solve Right Triangles?

Identify known sides or angles. Use SOH CAH TOA for trigonometric ratios:

Sine = opposite/hypotenuse
Tangent = opposite/adjacent Apply the Pythagorean theorem as needed.

## What Is the 45-45-90 Rule?

In a 45-45-90 triangle, the sides are in a ratio of 1:1:: √2​:

Both legs are equal.
Hypotenuse = leg x √2

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