Contents

- 1 Trigonometry Formulas
- 1.1 Trigonometry Formulas : Degree To Radian
- 1.2 Trigonometry Formulas : Radian To Degree
- 1.3 Trigonometry Formulas : Arc Length & Angle of Sector of Circle
- 1.4 Trigonometry Formulas : Pythagorean Theorem
- 1.5 Trigonometry Formulas : Trigonometric Ratios
- 1.6 Trigonometry Formulas : Quotient Identities
- 1.7 Trigonometry Formulas : Reciprocal Identities
- 1.8 Trigonometry Formulas : Trigonometric Identities
- 1.9 Trigonometry Formulas : Trigonometric Table
- 1.10 Trigonometry Formulas : Trigonometric Values
- 1.11 Trigonometry Formulas : Sign of Trigonometric Ratios
- 1.12 Trigonometry Formulas : Allied Angles Formula

# Trigonometry Formulas

Hi students, welcome to **Amans Maths Blogs (AMB)**. On this post, you will get the all the **Trigonometry Formulas**. It will help you to solve the trigonometry questions. This **Trigonometry Formulas **is very useful for in the revision before the school exams or competitive exams like SSC, IBPS, CAT, NTSE etc. This is in the form PDF file so you can download it your mobile/laptop and save this PDF file permanently.

## Trigonometry Formulas : Degree To Radian

To convert an angle in degree into radian, we need to multiply π/180 to the degree.

For example: 30 degree = 30 x π/180 = π/6 radian.

## Trigonometry Formulas : Radian To Degree

To convert an angle in radian into degree, we need to multiply 180/π to the degree.

For example: π/6 radian = π/6 x 180/π = 30 degree.

## Trigonometry Formulas : Arc Length & Angle of Sector of Circle

If an arc makes an angle θ (in radian) at the center of a circle whose radius is r, then the length of an arc of the arc is s = rθ. Hence, we can say that the angle θ = s/r radian.

## Trigonometry Formulas : Pythagorean Theorem

In a right angled triangle, the sum of the square of its perpendicular and the square of its base is equal to the square of the hypotenuse. This is known as **Pythagorean theorem**.

In the figure below, ABC is a right angled triangle with right angle at B. In this triangle, the side opposite to the right angle B is the hypotenuse (h = AC), the side opposite to the angle ACB = θ is the perpendicular (p = AB) and the side BC is the base (b).

Thus, according to the Pythagorean theorem p^{2} + b^{2} = h^{2}.

We can also write it as p^{2} = h^{2} – b^{2} or b^{2} = h^{2} – p^{2}.

## Trigonometry Formulas : Trigonometric Ratios

In trigonometry, we study the relation between the sides and angles of a right triangle. For this, six trigonometric ratios are defined as below.

1. Sine of θ is denoted as **sinθ** and it is the ratio of perpendicular (p) and hypotenuse (h). It means,

**sinθ = p/h = AB/AC.**

2. Cosine of θ is denoted as **cosθ** and it is the ratio of base (b) and hypotenuse (h). It means,

**cosθ = b/h = BC/AC.**

3. Tangent of θ is denoted as **tanθ** and it is the ratio of perpendicular (p) and base (b). It means,

**tanθ = p/b = AB/BC.**

4. Cosecant of θ is denoted as **cosecθ **or** cscθ** and it is the ratio of hypotenuse (h) and perpendicular (p). It means,

**cosecθ or cscθ = h/p = AC/AB.**

5. Secant of θ is denoted as **secθ** and it is the ratio of hypotenuse (h) and base (b). It means,

**secθ = h/b = AC/BC.**

6. Cotangent of θ is denoted as **cotθ** and it is the ratio of base (b) and hypotenuse (h). It means,

**cotθ = b/p = BC/AB.**

Read : Trigonometry Ratios Questions And Answer

## Trigonometry Formulas : Quotient Identities

Since sinθ = p/h, cosθ = b/h and tanθ = p/b, then tanθ = (p/h) / (b/h). Thus,

**tanθ = sinθ / cosθ**.

Since sinθ = p/h, cosθ = b/h and cotθ = b/p, then cotθ = (b/h) / (p/h). Thus,

**cotθ = cosθ / sinθ**.

## Trigonometry Formulas : Reciprocal Identities

Since sinθ = p/h and cosecθ = h/p, then

**sinθ x cosecθ = 1** or **sinθ = 1/ cosecθ** and **cosecθ = 1/ sinθ**.

Since cosθ = b/h and secθ = h/b, then

**cosθ x secθ = 1** or **cosθ = 1/ secθ** and **secθ = 1/ cosθ**.

Since tanθ = p/b and cotθ = b/p, then

**tanθ x cotθ = 1** or **tanθ = 1/ cotθ** and **cotθ = 1/ tanθ**.

## Trigonometry Formulas : Trigonometric Identities

Using Pythagorean theorem p^{2} + b^{2} = h^{2 }and the basic definition of trigonometric ratios, we get three trigonometric identities as below.

**sin ^{2}θ + cos^{2}θ = 1 **

**sec ^{2}θ – tan^{2}θ = 1 **

**cosec ^{2}θ – cot^{2}θ = 1**

Using a^{2} – b^{2} = (a + b)(a – b)^{ }in **sec ^{2}θ – tan^{2}θ = 1, **we get (secθ + tanθ)(secθ – tanθ) = 1. Thus, we get results as

**secθ + tanθ = 1 / (secθ – tanθ)**

and

**secθ – tanθ = 1 / (secθ + tanθ)**,

where θ nπ + π/2

Using a^{2} – b^{2} = (a + b)(a – b)^{ }in **cosec ^{2}θ – cot^{2}θ = 1, **we get (cosecθ + cotθ)(cosecθ – cotθ) = 1. Thus, we get results as

**cosecθ + cotθ = 1 / (cosecθ – cotθ)**

and

**cosecθ – cotθ = 1 / (cosecθ + cotθ)**,

where θ nπ.

Read : Trigonometry Identities Questions And Answer

## Trigonometry Formulas : Trigonometric Table

There are some specific trigonometric values of certain angles which are given below in trigonometric table.

Trigonometry Table | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Degree (θ)
Values |
0^{0} |
π/6 = 30^{0} |
π/4 = 45^{0} |
π/3 = 60^{0} |
π/2 = 90^{0} |
2π/3 = 120^{0} |
3π/4 = 135^{0} |
5π/6 = 150^{0} |
π = 180^{0} |
7π/6 = 210^{0} |
5π/4 = 225^{0} |
4π/3 = 240^{0} |
3π/2 = 270^{0} |
5π/3 = 300^{0} |
7π/4 = 315^{0} |
11π/6 = 330^{0} |
2π = 360^{0} |

sinθ |
0 | 1 | 0 | -1 | 0 | ||||||||||||

cosθ |
1 | 0 | -1 | 0 | 1 | ||||||||||||

tanθ |
0 | 1 | ∞ | -1 | 0 | 1 | ∞ | -1 | 0 | ||||||||

cosecθ |
∞ | 2 | 1 | 2 | ∞ | -2 | -1 | -2 | ∞ | ||||||||

secθ |
1 | 2 | ∞ | -2 | -1 | -2 | ∞ | 2 | 1 | ||||||||

cotθ |
∞ | 1 | 0 | -1 | ∞ | 1 | 0 | -1 | ∞ |

Read : Trigonometry Table Questions And Answer

## Trigonometry Formulas : Trigonometric Values

There are also trigonometric values of some important angles other than 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330 and 360 degrees as given in above trigonometric table.

1. sin15^{o} = cos75^{o} = ,

2. cos15^{o} = sin75^{o} = ,

3. tan15^{o} = cot75^{o} = ,

4. cot15^{o} = tan75^{o} = ,

5. sin22.5^{o} = cos67.5^{o} = ,

6. cos22.5^{o} = sin67.5^{o} = ,

7. tan22.5^{o} = ,

8. cot22.5^{o} = ,

9. sin18^{o} = cos72^{o} = ,

10. cos18^{o} = sin72^{o} = ,

11. sin36^{o} = cos54^{o} = ,

12. cos36^{o} = sin54^{o} = ,

## Trigonometry Formulas : Sign of Trigonometric Ratios

The sign of the trigonometric ratios (sinθ, cosθ, tanθ, cosecθ, secθ, cotθ) depends on the angle θ in which it lies in the quadrants. For this, we use the concept of **ASTC** rule.

**A** is at the **first** position (means **1 ^{st}** quadrant) in

**ASTC**which means All (

**A means All**). It means all the trigonometric ratios of an angle θ in first quadrant (0

^{o}< θ < 90

^{o}) are

**positive**.

**S** is at the **second** position (means **2 ^{nd}** quadrant) in

**ASTC**which means Sin and its reciprocal Cosec (

**S means Sin & Cosec**). It means only Sin and Cosec of an angle θ in second quadrant (90

^{o}< θ < 180

^{o}) are

**positive**and other remaining trigonometric ratios (cosθ, tanθ, secθ, cotθ) are negative.

**T** is at the **third** position (means **3 ^{rd}** quadrant) in

**ASTC**which means Tan and its reciprocal Cot (

**T means Tan & Cot**). It means only Tan and Cot of an angle θ in third quadrant (180

^{o}< θ < 270

^{o}) are

**positive**and other remaining trigonometric ratios (sinθ, cosθ, cosecθ, secθ) are negative.

**C** is at the **fourth** position (means **4 ^{th}** quadrant) in

**ASTC**which means Cos and its reciprocal Sec (

**C means Cos & Sec**). It means only Cos and Sec of an angle θ in fourth quadrant (270

^{o}< θ < 360

^{o}) are

**positive**and other remaining trigonometric ratios (sinθ, tanθ, cosecθ, cotθ) are negative.

## Trigonometry Formulas : Allied Angles Formula

The angles **90 ^{o} θ**,

**180**,

^{o}θ**270**and

^{o}θ**360**are knwon as

^{o}θ**allied angles**. The value of trigonometric ratios of these allied angles is according to the ASTC rule as discussed above.

Here, another rule is also used known as the **odd and even multiplications of 90 ^{o}**.

**90 ^{o }**= 1 x

**90**and

^{o }**270**= 3 x

^{o }**90**are the odd multiplication of

^{o}**90**. In this case, the trigonometric ratios are changed in pairs. It means

^{o}**sin is changed to cos and cos is changed to sin**

**tan is changed to cot and cot is changed to tan**

**cosec is changed to sec and sec is changed to cosec**

**180 ^{o }**= 2 x

**90**and

^{o }**360**= 4 x

^{o }**90**are the even multiplication of

^{o}**90**. In this case, the trigonometric ratios are

^{o}**NOT**changed. It means

**sin remains sin and cos remains cos**

**tan remains tan and cot remains cot**

**cosec remains cosec and sec remains sec**

Now, the allied angle **(90 ^{o} – θ)** is in the first quadrant and

**90**= 1 x

^{o }**90**is odd multiplicative of

^{o }**90**, so

^{o}**all the trigonometric ratios of (90**and all the trigonometric ratios are changed according to discussed above.

^{o}– θ) will be positiveHence, we conclude that

**sin(90 ^{o} – θ) = +cosθ**

**cos(90 ^{o} – θ) = +sinθ**

**tan(90 ^{o} – θ) = +cotθ**

**cosec(90 ^{o} – θ) = +secθ**

**sec(90 ^{o} – θ) = +cosecθ**

**cot(90 ^{o} – θ) = +tanθ**

The allied angle **(90 ^{o} + θ)** is in the second quadrant and

**90**= 1 x

^{o }**90**is odd multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**sin and cosec of (90**and other will be negative and all the trigonometric ratios are changed in pairs.

^{o}+ θ) will be positiveHence, we conclude that

**sin(90 ^{o} + θ) = +cosθ**

**cos(90 ^{o} + θ) = -sinθ**

**tan(90 ^{o} + θ) = -cotθ**

**cosec(90 ^{o} + θ) = +secθ**

**sec(90 ^{o} + θ) = -cosecθ**

**cot(90 ^{o} + θ) = -tanθ**

The allied angle **(180 ^{o} – θ)** is in the second quadrant and

**180**= 2 x

^{o }**90**is even multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**sin and cosec of (180**and other will be negative and all the trigonometric ratios remain same.

^{o}– θ) will be positiveHence, we conclude that

**sin(180 ^{o} – θ) = +sinθ**

**cos(180 ^{o} – θ) = -cosθ**

**tan(180 ^{o} – θ) = -tanθ**

**cosec(180 ^{o} – θ) = +cosecθ**

**sec(180 ^{o} – θ) = -secθ**

**cot(180 ^{o} – θ) = -cotθ**

The allied angle **(180 ^{o} + θ)** is in the third quadrant and

**180**= 2 x

^{o }**90**is even multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**tan and cot of (180**and other will be negative and all the trigonometric ratios remain same.

^{o}+ θ) will be positiveHence, we conclude that

**sin(180 ^{o} + θ) = -sinθ**

**cos(180 ^{o} + θ) = -cosθ**

**tan(180 ^{o} + θ) = +tanθ**

**cosec(180 ^{o} + θ) = -cosecθ**

**sec(180 ^{o} + θ) = -secθ**

**cot(180 ^{o} + θ) = +cotθ**

The allied angle **(270 ^{o} – θ)** is in the third quadrant and

**270**= 3 x

^{o }**90**is odd multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**tan and cot of (270**and other will be negative and all the trigonometric ratios are changed in pairs.

^{o}– θ) will be positiveHence, we conclude that

**sin(270 ^{o} – θ) = -cosθ**

**cos(270 ^{o} – θ) = -sinθ**

**tan(270 ^{o} – θ) = +cotθ**

**cosec(270 ^{o} – θ) = -secθ**

**sec(270 ^{o} – θ) = -cosecθ**

**cot(270 ^{o} – θ) = +tanθ**

The allied angle **(270 ^{o} + θ)** is in the fourth quadrant and

**270**= 3 x

^{o }**90**is odd multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**cos and sec of (270**and other will be negative and all the trigonometric ratios are changed in pairs.

^{o}+ θ) will be positiveHence, we conclude that

**sin(270 ^{o} + θ) = -cosθ**

**cos(270 ^{o} + θ) = +sinθ**

**tan(270 ^{o} + θ) = -cotθ**

**cosec(270 ^{o} + θ) = -secθ**

**sec(270 ^{o} + θ) = +cosecθ**

**cot(270 ^{o} + θ) = -tanθ**

The allied angle **(360 ^{o} – θ)** is in the fourth quadrant and

**360**= 4 x

^{o }**90**is even multiplicative of

^{o }**90**, so the trigonometric ratios

^{o}**cos and sec of (360**and other will be negative and all the trigonometric ratios remain same.

^{o}– θ) will be positiveHence, we conclude that

**sin(360 ^{o} – θ) = -sinθ**

**cos(360 ^{o} – θ) = +cosθ**

**tan(360 ^{o} – θ) = -tanθ**

**cosec(360 ^{o} – θ) = -cosecθ**

**sec(360 ^{o} – θ) = +secθ**

**cot(360 ^{o} – θ) = -cotθ**

The allied angle **(360 ^{o} + θ)** is again in the first quadrant and

**360**= 4 x

^{o }**90**is even multiplicative of

^{o }**90**, so

^{o}**all the trigonometric ratios of (360**and other will be negative and all the trigonometric ratios remain same.

^{o}+ θ) will be positiveHence, we conclude that

**sin(360 ^{o} + θ) = +sinθ**

**cos(360 ^{o} + θ) = +cosθ**

**tan(360 ^{o} + θ) = +tanθ**

**cosec(360 ^{o} + θ) = +cosecθ**

**sec(360 ^{o} + θ) = +secθ**

**cot(360 ^{o} + θ) = +cotθ**