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December 25, 201539 Insightful Publications

**Trigonometry Formulas**: Trigonometry is the branch of Mathematics. It deals with the relationship between a triangle’s sides and angles. The students can learn basic trigonometry formulas and concepts from textbooks. Also, they can learn its application to daily life things, such as, if you are standing on the terrace of a tall building at a given height and see a post box on the other side of the road, you can easily calculate the width of the road using trigonometry.

Students are taught the basic concepts regarding Trigonometry from Class 10. These trigonometry formulas are very helpful in astronomy to calculate the distance between stars and satellites. In this article, students shall learn about all trigonometry formulas, their representation as ratio tables, how to measure the sides of angles, calculate trigonometry values, and determine the distance between landmarks.

There are six fundamental trigonometric ratios used in all formulas of trigonometry. These ratios are also known as trigonometric functions and mostly use all trigonometry formulas. The six essential trigonometric functions are sine, cosine, secant, cosecant, tangent, and cotangent.

The trigonometric functions and identities are derived by using the right-angled triangle. When the height and the base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas.

Trigonometry Formula is the branch of Maths that deals primarily with triangles. It is also called the study of the relationships between the lengths and angles of a triangle. When learning about trigonometric formulas, we need to consider only right-angled triangles. However, they can be applied to other triangles also.

In a right-angled triangle, there are three sides: hypotenuse, the opposite side (perpendicular), and the adjacent side (base). The longest side is called the hypotenuse. The side opposite to the angle is the perpendicular, and the side where both the hypotenuse and opposite side rests is the adjacent side.

Various sets of formulas for trigonometry are given below:

- Basic Formulas
- Reciprocal Identities
- Trigonometric Ratio Table
- Periodic Identities
- Cofunction Identities
- Sum and Difference of Identities
- Half-Angle Identities
- Double Angle Identities
- Triple Angle Identities
- Product Identities
- Sum of Product Identities
- Inverse Trigonometry Formulas
- Sine Law and Cosine Law

Below we have provided the list of basic trigonometry formulas for your reference:

- \( \sin \theta = \frac{{{\rm{ Opposite\, Side }}}}{{{\rm{ Hypotenuse }}}}\)
- \( \cos \theta = \frac{{{\rm{ Adjacent\, Side }}}}{{{\rm{ Hypotenuse }}}}\)
- \( \tan \theta = \frac{{{\rm{ Opposite\, Side }}}}{{{\rm{ Adjacent Side }}}}\)
- \( \sec \theta = \frac{{{\rm{ Hypotenuse }}}}{{{\rm{ Adjacent\, Side }}}}\)
- \({\rm{cosec}}\,\theta = \frac{{{\rm{ Hypotenuse }}}}{{{\rm{ Opposite\, Side }}}}\)
- \( \cot \theta = \frac{{{\rm{ Adjacent\, Side }}}}{{{\rm{ Opposite\, Side }}}}\)

You can also save the image below for easy reference:

Cosecant, secant, and cotangent are the reciprocals of the basic trigonometric ratios sine, cosine, and tangent. All of these common identities are also taken from the right-angled triangle. The reciprocal trigonometric identities are taken by using the trigonometric functions. They are utilised frequently to simplify trigonometric problems.

- \({\rm{cosec}}\,\theta = \frac{1}{{ \sin \theta }}\)
- \( \sec \theta = \frac{1}{{ \cos \theta }}\)
- \( \cot \theta = \frac{1}{{ \tan \theta }}\)
- \( \sin \theta = \frac{1}{\theta }\)
- \( \cos \theta = \frac{1}{{ \sec \theta }}\)
- \( \tan \theta = \frac{1}{{ \cot \theta }}\)

- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(\cot^2 \theta + 1 = {\rm{cosec}}^2 \theta \)
- \(1 + \tan^2 \theta = \sec^2 \theta \)

The below trigonometry table formula shows all trigonometry formulas and commonly used angles for solving trigonometric problems. The trigonometric ratios table helps find the values of standard trigonometric angles like \({0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ }\) and \({90^ \circ }.\)

Angles (In Degrees) | \({0^ \circ }\) | \({30^ \circ }\) | \({45^ \circ }\) | \({60^ \circ }\) | \({90^ \circ }\) | \({180^ \circ }\) | \({270^ \circ }\) | \({360^ \circ }\) |

Angles (In Radians) | \({0^ \circ }\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\pi\) | \(\frac{3 \pi}{2}\) | \(2 \pi\) |

\(\sin \) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(0\) | \(-1\) | \(0\) |

\(\cos \) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) | \(-1\) | \(0\) | \(1\) |

\(\tan \) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | \(\infty \) | \(0\) | \(\infty \) | \(0\) |

\(\cot \) | \(\infty \) | \(\sqrt{3}\) | \(1\) | \(\frac{1}{\sqrt{3}}\) | \(0\) | \(\infty \) | \(0\) | \(\infty \) |

\({\rm{cosec}}\) | \(\infty \) | \(2\) | \(\sqrt{2}\) | \(\frac{2}{{\sqrt 3 }}\) | \(1\) | \(\infty \) | \(-1\) | \(\infty \) |

\(\sec \) | \(1\) | \(\frac{2}{{\sqrt 3 }}\) | \(\sqrt{2}\) | \(2\) | \(\infty \) | \(-1\) | \(\infty \) | \(1\) |

**Learn About Trigonometric Values Table**

Periodicity formulas or identities are utilised to shift the angles by \(\frac{\pi }{2},\pi \), and \(2\pi \) The periodicity identities are also termed the co-function identities. All the trigonometric identities are cyclic, which means they repeat themselves after a period. The period differs for various trigonometric identities.

**Example:** \(\tan {45^ \circ } = \tan {225^ \circ },\) but the similar one is not true for \(\cos {45^ \circ }\) and \(\cos {225^ \circ }.\)

\(\sin \left( {\frac{\pi }{2} – A} \right) = \cos A\) | \(\cos \left( {\frac{\pi }{2} – A} \right) = \sin A\) |

\(\sin \left( {\frac{\pi }{2} + A} \right) = \cos A\) | \(\cos \left( {\frac{\pi }{2} + A} \right) = – \sin A\) |

\(\tan \left( {\frac{\pi }{2} + A} \right) = A\) | \(\cot \left( {\frac{\pi }{2} + A} \right) = – \tan A\) |

\(\tan \left( {\frac{\pi }{2} – A} \right) = \cot A\) | \(\cot \left( {\frac{\pi }{2} – A} \right) = \tan A\) |

\(\sin (\pi – A) = \sin A\) | \(\cos (\pi – A) = – \cos A\) |

\(\sin (\pi + A) = \sin A\) | \(\cos (\pi + A) = – \cos A\) |

\(\tan (\pi + A) = \tan A\) | \(\cot (\pi + A) = \tan A\) |

\(\tan (\pi – A) = – \tan A\) | \(\cot (\pi – A) = – \cot A\) |

\(\sin \left( {\frac{{3\pi }}{2} – A} \right) = – \cos A\) | \(\cos \left( {\frac{{3\pi }}{2} – A} \right) = – \sin A\) |

\(\sin \left( {\frac{{3\pi }}{2} + A} \right) = – \cos A\) | \(\cos \left( {\frac{{3\pi }}{2} + A} \right) = \sin A\) |

\(\tan \left( {\frac{{3\pi }}{2} + A} \right) = A\) | \(\cot \left({\frac{{3\pi}}{2} + A} \right) = – \tan A\) |

\(\tan \left( {\frac{{3\pi }}{2} – A} \right) = \cot A\) | \(\cot \left( {\frac{{3\pi }}{2} – A} \right) = \tan A\) |

\(\sin (2\pi – A) = \sin A\) | \(\cos (2\pi – A) = \cos A\) |

\(\sin (2\pi + A) = \sin A\) | \(\cos (2\pi + A) = – \cos A\) |

\(\tan (2\pi + A) = \tan A\) | \(\cot (2\pi + A) = \cot A\) |

\(\tan (2\pi – A) = – \tan A\) | \(\cot (2\pi – A) = – \cot A\) |

The cofunction identities provide the interrelationship between the different trigonometric functions. The cofunction (periodic identities) are shown in the degrees below:

- \( \cos \left( {{{90}^ \circ } – x} \right) = \sin x\)
- \(\tan \left( {{{90}^ \circ } – x} \right) = \cot x\)
- \( \cot \left( {{{90}^ \circ } – x} \right) = \tan x\)
- \(\sec \left( {{{90}^ \circ } – x} \right) = {\rm{cosec}}\,x\)
- \({\rm{cosec}}\,\left( {{{90}^ \circ } – x} \right) = \sec x\)

The formulas of the sum and difference identities include

- \( \sin (x + y) = \sin x \cos y + \cos x \sin y\)
- \( \cos (x + y) = \cos x \cos y – \sin x \sin y\)
- \( \tan (x + y) = \frac{{( \tan x + \tan y)}}{{(1 – \tan x \cdot \tan y)}}\)
- \( \sin (x – y) = \sin x \cos y – \cos x \sin y\)
- \( \cos (x – y) = \cos x \cos y + \sin x \sin y\)
- \( \tan (x – y) = \frac{{x – \tan y)}}{{(1 + \tan x \cdot \tan y)}}\)

- \( \sin \frac{x}{2} = \pm \sqrt {\frac{{1 – \cos x}}{2}}\)
- \( \cos \frac{x}{2} = \pm \sqrt {\frac{{1 + \cos x}}{2}} \)
- \( \tan {\frac{x}{2}} = {\frac{1 – \cos x}{\sin x}} = {\frac{\sin x}{1+\cos x}}= \pm \sqrt {\frac{{1 – \cos x}}{{1 + \cos x}}} \)
- \(\cot {\frac{x}{2}} = {\frac{\sin x}{1 – \cos x}} = {\frac{1+\cos x}{\sin x}}= \pm \sqrt {\frac{{1 + \cos x}}{{1 – \cos x}}} \)

The double of the angle \(x\) is represented by the given formulas**:**

- \( \sin 2x = 2 \sin x \cos x = {\frac{{2 \tan x}}{{1 + \tan^2 x}}}\)
- \( \cos 2x = – x = \cos^2 x – \sin^2 x = 2 \cos^2 x – 1 = 1 – 2 \sin^2 x = {\frac{1 – \tan^2 x}{1+\tan^2 x}}\)
- \( \tan 2x = \frac{{2\tan x}}{{1 – \tan^2 x}}\)
- \( \cot 2x = \frac{{\cot^2 x – 1}}{{2 \cot x}}\)
- \( \sec 2x = \frac{\sec^2 x}{{(2 – \sec^2 x)}}\)
- \({\rm{cosec}}\,2x = \frac{\sec x \cdot {\rm{cosec}}\,x}{2}\)

- \(\sin 3x = – {\sin^3}x + 3{\cos^2}x\sin x = – 4{\sin^3}x + 3\sin x\)
- \(\cos 3x = – {\cos^3}x + 3{\sin^2}x\cos x = 4{\cos^3}x – 3\cos x\)
- \(\tan 3x = \frac{{3\tan x – {{\tan }^3}x}}{{1 – 3{{\tan }^2}x}}\)
- \(\cot 3x = \frac{{3\cot x – {{\cot }^3}x}}{{1 – 3{{\cot }^2}x}}\)

- \(\cos \theta \,\cos \varphi = \frac{{\cos (\theta – \varphi ) + \cos (\theta + \varphi )}}{2}\)
- \(\sin \theta \,\sin \varphi = \frac{{\cos (\theta – \varphi ) – \cos (\theta + \varphi )}}{2}\)
- \(\sin \theta \,\cos \varphi = \frac{{\sin (\theta + \varphi ) + \sin (\theta – \varphi )}}{2}\)
- \(\cos \theta \,\sin \varphi = \frac{{\sin (\theta + \varphi ) – \sin (\theta – \varphi )}}{2}\)
- \(\tan \theta \,\tan \varphi = \frac{{\cos (\theta – \varphi ) – \cos (\theta + \varphi )}}{{\cos (\theta – \varphi ) + \cos (\theta + \varphi )}}\)

The combination of two of the acute angles \(A\) and \(B,\) can be represented through the trigonometric ratios, as shown below:

- \(\sin \theta \pm \sin \varphi = 2\sin \left( {\frac{{\theta \pm \varphi }}{2}} \right)\cos \left( {\frac{{\theta \mp \varphi }}{2}} \right)\)
- \(\cos \theta + \cos \varphi = 2\cos \left( {\frac{{\theta + \varphi }}{2}} \right)\cos \left( {\frac{{\theta – \varphi }}{2}} \right)\)
- \(\cos \theta – \cos \varphi = – 2\sin \left( {\frac{{\theta + \varphi }}{2}} \right)\sin \left( {\frac{{\theta – \varphi }}{2}} \right)\)
- \(\tan \theta \pm \tan \varphi = \frac{{\sin (\theta \pm \varphi )}}{{\cos \theta \cos \varphi }}\)

The ratios of trigonometry are inverted to create the inverse trigonometric functions. \(\sin \theta = x\) and \(\theta = \sin^{-1} x\) . So, \(x\) can have the values in whole numbers, decimals, fractions or exponents.

- \({\sin ^{ – 1}}( – x) = – {\sin ^{ – 1}}x,\,x \in [ – 1,\,1]\)
- \({\tan ^{ – 1}}( – x) = – {\tan ^{ – 1}}x,\,x \in R\)
- \({\operatorname{cosec} ^{ – 1}}( – x) = – {\operatorname{cosec} ^{ – 1}}x,\,x \in R – ( – 1,\,1)\)
- \({\cos ^{ – 1}}( – x) = \pi – {\cos ^{ – 1}}x,\,x \in [ – 1,\,1]\)
- \({\sec ^{ – 1}}( – x) = \pi – {\sec ^{ – 1}}x,\,x \in R – ( – 1,\,1)\)
- \({\cot ^{ – 1}}( – x) = \pi – {\cot ^{ – 1}}x,\,x \in R\)

**Sine laws**: The sine law and the cosine law give us the relationship between the sides and the angles of a triangle. The Sine Law gives the ratio of the sides and the angle opposite to the side.

**Example:** The ratio can be taken for the side a and its opposite angle ‘A’.\( = \frac{{ \sin B}}{b} = \frac{{ \sin C}}{c}\)

**Cosine laws**: The cosine Law helps to find the length of a side for the given lengths of the other two sides and the included angle. For example, the length \(‘a’\) can be found with the help of sides \(b\) and \(c,\) and their included angle \(A.\)

\({a^2} = {b^2} + {c^2} – 2bc \cos A\)

\({b^2} = {a^2} + {c^2} – 2ac \cos B\)

\({c^2} = {a^2} + {b^2} – 2ab\cos C\)

\(a, b\) and \(c\) are the lengths of sides of the triangle, and \(A, B, C\) are the angles of the triangle.

**Q.1. Sam has to find the value of \(\sin15°\) by using the trigonometry formulas. How can you help Sam to see the value?Ans:** Given: \(\sin15^{\circ}\)

\(=\sin (45^{\circ} – 30^{\circ})\)

\(=\sin 45^{\circ} ⋅ \cos 30^{\circ} – \cos 45^{\circ} ⋅ \sin 30^{\circ}\)

\( = \frac{1}{{\surd 2}} \times \frac{{\surd 3}}{2} – \frac{1}{{\sqrt 2 }} \times \frac{1}{2} = \frac{{\sqrt 3 – 1}}{{2\surd 2}}\)

Hence, the required answer is \(\sin 15^{\circ} = \frac{{\surd 3}-1}{2\surd 2}\)

**Q.2. If \(\cos A = {\frac{4}{5}}\) then \(\tan A =\)?Ans:** Given,

\(\cos A = \frac{4}{5}\)

As we know, the trigonometry identities,

\(1+\tan^2 A = \sec^2 A\)

\(\sec^2 A – 1 = \tan^2 A\)

\((\frac{1}{\cos^2 A}) – 1 = \tan^2 A\)

Putting the value of \(\cos A = \frac{4}{5}\)

\((\frac{5}{4})^2 – 1 = \tan^2 A\)

\(\tan^2 A = \frac{9}{16}\)

\(\tan A = \frac{3}{4}\)

Hence, the required answer is \(\tan A = \frac{3}{4}\)

**Q.3. If \(\sin θ⋅ \cos θ = 5\) find the value of \((\sin θ+\cos θ)^2\), using the trigonometry formulas.Ans:** \((\sin θ+\cos θ)^2\)

\(=\sin^2 θ + \cos^2 θ+2 \sin θ ⋅ \cos θ\)

\(=(1)+2(5)=1+10=11\)

\((\sin θ+\cos θ)^2=11\)

Hence, the required answer is \(11\).

**Q.4. Ria is given the trigonometric ratio of \(\tan θ = \frac{5}{12}\). Can you help Ria find the trigonometric ratio of \({\rm{cosec}}\,θ\) using the formulas.Ans:** \(\tan θ = \frac{{\text{Perpendicular}}}{{\text{Base}}} = \frac{5}{12}\)

Perpendicular \(=5\) and Base \(=12\)

\({\rm{Hypotenuse}}^2 = {\rm{Perpendicular}}^2 + {\rm{Base}}^2\)

\(⇒ {\rm{Hypotenuse}}^2 = 5^2 + 12^2\)

\(⇒ {\rm{Hypotenuse}}^2= 25+144\)

\(⇒ {\rm{Hypotenuse}} = \sqrt {169}\)

\(⇒ {\rm{Hypotenuse}} =13\)

Hence, \(\sin θ = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{5}{13}\)

\({\text{cosec}}\,θ = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{13}{5}\).

**Q.5. What is the value of \((\sin 30^{\circ}+\cos 30^{\circ})–(\sin 60^{\circ}+\cos 60^{\circ})\)?Ans:** Given,

\((\sin 30^{\circ}+\cos 30^{\circ})–(\sin 60^{\circ}+\cos 60^{\circ})\)

\( = \frac{1}{2} + \frac{{\surd 3}}{2} – \frac{{\sqrt 3 }}{2} – \frac{1}{2} = 0\)

Hence, the required answer is \(0\).

**Ans:** The eleven trigonometric identities in trigonometry are given below:

Basic Formulas

Reciprocal Identities

Trigonometric Ratio Table

Periodic Identities

Cofunction Identities

Sum and Difference Identities

Half-Angle Identities

Double Angle Identities

Triple Angle Identities

Product Identities

Sum of Product Identities

**Ans:** Sine, cosine, tangent, contangent, secant and cosecant are the basic trigonometric ratios.

**Ans: **The three main functions in trigonometry are Sin, Cos and Tan.

**Q.4. What are the three Pythagorean identities?****Ans: **The first identity states that Sine squared plus cosine squared identical one. The second one states that tangent squared plus one is similar to Secant squared. For the last one, it says that one plus cotangent squared is comparable to cosecant squared.

Formulas are:

\({\sin ^2}\theta + {\cos ^2}\theta = 1\)

\({\cot ^2}\theta + 1 = {{\text{cosec}}^2}\theta \)

\(1 + {\tan ^2}\theta = {\sec ^2}\theta \)

**Q.5. What are the three formulas of trigonometry?****Ans:** The three formulas of trigonometry are Sine, cosine and tangent. The formulas are given below:

**Sine Function:** \(\sin \theta = \frac{{{\text{ Opposite Side }}}}{{{\text{ Hypotenuse }}}}\)**Cosine Function: **\(\cos \theta = \frac{{{\text{ Adjacent Side }}}}{{{\text{ Hypotenuse }}}}\)**Tangent Function:** \(\tan \theta = \frac{{{\text{ Opposite Side }}}}{{{\text{ Adjacent Side }}}}\)