Different Rules and Formulas Relating the Sides and Angles of the Triangle: Rules, Formulas

May 20, 202239 Insightful Publication

**Trigonometry Formulas**: Trigonometry is the branch of Mathematics. This section deals with the relationship between a triangle’s sides and angles. The students can learn trigonometry concepts and formulas from textbooks. Also, they can learn its application to daily life things such as, if you’re 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 quickly calculate the width of the road using trigonometry.

These trigonometry formulas are very helpful in astronomy to calculate the distance between stars and satellites. In this article, students shall learn about 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 of trigonometry are given below:

1. Basic Formulas

2. Reciprocal Identities

3. Trigonometric Ratio Table

4. Periodic Identities

5. Cofunction Identities

6. Sum and Difference of Identities

7. Half-Angle Identities

8. Double Angle Identities

9. Triple Angle Identities

10. Product Identities

11. Sum of Product Identities

12. Inverse Trigonometry Formulas

13. Sine Law and Cosine Law

1. \( \sin \theta = \frac{{{\rm{ Opposite\, Side }}}}{{{\rm{ Hypotenuse }}}}\)

2. \( \cos \theta = \frac{{{\rm{ Adjacent\, Side }}}}{{{\rm{ Hypotenuse }}}}\)

3. \( \tan \theta = \frac{{{\rm{ Opposite\, Side }}}}{{{\rm{ Adjacent Side }}}}\)

4. \( \sec \theta = \frac{{{\rm{ Hypotenuse }}}}{{{\rm{ Adjacent\, Side }}}}\)

5. \(\theta = \frac{{{\rm{ Hypotenuse }}}}{{{\rm{ Opposite\, Side }}}}\)

6. \( \cot \theta = \frac{{{\rm{ Adjacent\, Side }}}}{{{\rm{ Opposite\, Side }}}}\)

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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.

(i) \(\theta = \frac{1}{{ \sin \theta }}\)

(ii) \( \sec \theta = \frac{1}{{ \cos \theta }}\)

(iii) \( \cot \theta = \frac{1}{{ \tan \theta }}\)

(iv) \( \sin \theta = \frac{1}{\theta }\)

(v) \( \cos \theta = \frac{1}{{ \sec \theta }}\)

(vi) \( \tan \theta = \frac{1}{{ \cot \theta }}\)

(i) \(\theta + \theta = 1\)

(ii) \(\theta + 1 = \theta \)

(iii) \(1 + \theta = \theta \)

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 true for \(\cos {45^ \circ }\) and \(\cos {225^ \circ }.\)

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

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

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

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

(v) \( \sin (\pi – A) = \sin A\& \cos (\pi – A) = – \cos A\)

(vi) \( \sin (\pi + A) = \sin A\& \cos (\pi + A) = – \cos A\)

(vii) \( \tan (\pi + A) = \tan A\& \cot (\pi + A) = \tan A\)

(viii) \( \tan (\pi – A) = – \tan A\& \cot (\pi – A) = – \cot A\)

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

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

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

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

(xiii) \( \sin (2\pi – A) = \sin A\& \cos (2\pi – A) = \cos A\)

(xiv) \( \sin (2\pi + A) = \sin A\& \cos (2\pi + A) = – \cos A\)

(xv) \( \tan (2\pi + A) = \tan A\& \cot (2\pi + A) = \cot A\)

(xvi) \( \tan (2\pi – A) = – \tan A\& \cot (2\pi – A) = – \cot A\)

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

(i) \( \cos \left( {{{90}^ \circ } – x} \right) = \sin x\)

(ii) \(\tan \left( {{{90}^ \circ } – x} \right) = \cot x\)

(iii) \( \cot \left( {{{90}^ \circ } – x} \right) = \tan x\)

(iv) \(\sec \left( {{{90}^ \circ } – x} \right) = x\)

(v) \(\left( {{{90}^ \circ } – x} \right) = \sec x\)

The formulas of the sum and difference identities include

(i) \( \sin (x + y), \cos (x – y), \cot (x + y),etc.\)

(ii) \( \sin (x + y) = \sin (x)\cos (y) + \cos (x) \sin (y)\)

(iii) \( \cos (x + y) = \cos (x) \cos (y) – \sin (x) \sin (y)\)

(iv) \( \tan (x + y) = \frac{{( \tan x + \tan y)}}{{(1 – \tan x \cdot \tan y)}}\)

(v) \( \sin (x – y) = \sin (x) \cos (y) – \cos (x) \sin (y)\)

(vi) \( \cos (x – y) = \cos (x) \cos (y) + \sin (x) \sin (y)\)

(vii) \( \tan (x – y) = \frac{{x – \tan y)}}{{(1 + \tan x \cdot \tan y)}}\)

(i) \( \sin \frac{x}{2} = \pm \sqrt {\frac{{1 – \cos x}}{2}}\)

(ii) \( \cos \frac{x}{2} = \pm \sqrt {\frac{{1 + \cos x}}{2}} \)

(iii) \( \tan \left( {\frac{x}{2}} \right) = \sqrt {\frac{{1 – \cos (x)}}{{1 + \cos (x)}}} \)

(iv) \(\left( {\frac{x}{2}} \right) = \sqrt {\frac{{1 – \cos (x)}}{{1 + \cos (x)}}} \)

(v) \( = \frac{{1 – \cos (x)}}{{ \sin (x)}}\)

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

(i) \( \sin (2x) = (x) \cdot \cos (x) = \left[ {\frac{{2 \tan x}}{{(1 + x)}}} \right]\)

(ii) \( \cos (2x) = – (x) = \left[ {\frac{{(1 – x)}}{{(1 + x)}}} \right]\)

(iii) \( \cos (2x) = 2(x) – 1 = 1 – 2(x)\)

(iv) \( \tan (2x) = \frac{{[2\tan (x)]}}{{[1 – (x)]}}\)

(v) \( \sec (2x) = \frac{x}{{(2 – x)}}\)

(vi) \((2x) = \frac{{ \sec x \times x}}{2}\)

**Practice 10th CBSE Exam Questions**

(i) \(\sin 3x = 3 \sin x – 4x\)

(ii) \( \cos 3x = 4x – 3 \cos x\)

(iii) \(\tan 3x = \frac{{[3 \tan x – x]}}{{[1 – 3x]}}\)

(i) \( \sin x \cdot \cos y = \frac{{(x + y) + \sin (x – y)]}}{2}\)

(ii) \(\cos x \cdot \cos y = \frac{{[\cos (x + y) + \cos (x – y)]}}{2}\)

(iii) \( \sin x \cdot \sin y = \frac{{[ \cos (x – y) – \cos (x + y)]}}{2}\)

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

(i) \( \sin x + \sin y = 2\left[ { \sin \frac{{x + y}}{2} \cos \frac{{x – y}}{2}} \right]\)

(ii) \( \sin x – \sin y = 2\left[ { \cos \frac{{x + y}}{2} \sin \frac{{x – y}}{2}} \right]\)

(iii) \( \cos x + \cos y = 2\left[ {\cos \frac{{x + y}}{2} \cos \frac{{x – y}}{2}} \right]\)

(iv) \( \cos x – \cos y = – 2\left[ { \sin \frac{{x + y}}{2} \sin \frac{{x – y}}{2}} \right]\)

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

(i) \(( – x) = – x\)

(ii) \(( – x) = x = \pi – x\)

(iii) \(( – x) = – x\)

(iv) \(( – x) = – x\)

(v) \(( – x) = \pi – x\)

(vi) \(( – x) = \pi – x\)

**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 \(sin15°sin15°\) by using the trigonometry formulas. How can you help Sam to see the value?** 15∘15∘

Ans:

=(45∘–30∘)=(45∘–30∘)

=45∘⋅30∘–45∘⋅30∘=45∘⋅30∘–45∘⋅30∘

=12√×3√2–12√×12=3√–122√=12×32–12×12=3–122

Hence, the required answer is 15∘=3√–122√15∘=3–122

** Q.2. If cosA=45cosA=45 then tanA=tanA= ?** Given,

Ans:

cosA=45cosA=45

As we know, the trigonometry identities,

1+A=A1+A=A

A–1=AA–1=A

(1A)–1=A(1A)–1=A

Putting the value of cosA=45cosA=45

(54)2–1=A(54)2–1=A

A=916A=916

tanA=34tanA=34

Hence, the required answer is tanA=34tanA=34

** Q.3. If sinθ⋅cosθ=5sinθ⋅cosθ=5 find the value of (sinθ+cosθ)2(sinθ+cosθ)2, using the trigonometry formulas.** (sinθ+cosθ)2(sinθ+cosθ)2

Ans:

=θ+θ+2sinθ⋅cosθ=θ+θ+2sinθ⋅cosθ

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

(sinθ+cosθ)2=11(sinθ+cosθ)2=11

Hence, the required answer is 11.11.

** Q.4. Ria is given the trigonometric ratio of tanθ=512tanθ=512 Can you help Ria find the trigonometric ratio of θθ using the formulas.** tanθ=PerpendicularBase=512tanθ=PerpendicularBase=512

Ans:

Perpendicular=5andBase=12Perpendicular=5andBase=12

Hypotenuse2=Perpendicular2+Base2Hypotenuse2=Perpendicular2+Base2

⇒Hypotenuse2=52+122⇒Hypotenuse2=52+122

⇒Hypotenuse2=25+144⇒Hypotenuse2=25+144

⇒Hypotenuse=169−−−√⇒Hypotenuse=169

⇒Hypotenuse=13⇒Hypotenuse=13

Hence sinsinθ=PerpendicularHypotenuse=513sinsinθ=PerpendicularHypotenuse=513

θ=HypotenusePerpendicular=135θ=HypotenusePerpendicular=135

** Q.5. What is the value of (sin30∘+cos30∘)–(sin60∘+cos60∘)(sin30∘+cos30∘)–(sin60∘+cos60∘)?** Given,

Ans:

(sin30∘+cos30∘)–(sin60∘+cos60∘)(sin30∘+cos30∘)–(sin60∘+cos60∘)

=12+3√2–3√2–12=0=12+32–32–12=0

Hence, the required answer is 00.

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

Ans:

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

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

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

At Embibe, you can learn Trigonometry Formula Class 11, 12, 10 and practice trigonometry questions of Class 10, 11, and 12 for free. Embibe provides you with an incredible opportunity. Check out the table below for Embibe’s resources to master Trigonometry:

**Attempt 10th CBSE Exam Mock Tests**

** Q What are the 11 trigonometric identities?** The eleven trigonometric identities in trigonometry are given below:

Ans:

1. Basic formulas

2. Reciprocal Identities

3. Trigonometric Ratio Table

4. Periodic Identities

5. Cofunction Identities

6. Sum and Difference Identities

7. Half-Angle Identities

8. Double Angle Identities

9. Triple Angle Identities

10. Product Identities

11. Sum of Product Identities

** Q What are three Pythagorean identities?** 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.

Ans:

Formulas are:

θ+θ=1θ+θ=1

θ+1=θθ+1=θ

1+θ=θ1+θ=θ

** Q What are the three formulas of trigonometry?**The three formulas of trigonometry are Sine, cosine and tangent.

Ans:

The formulas are given below:

Sine function: sin(θ)=OppositeHypotenusesin(θ)=OppositeHypotenuse

Cosine Function: cos(θ)=AdjacentHypotenusecos(θ)=AdjacentHypotenuse

Tangent Function: tan(θ)=OppositeAdjacenttan(θ)=OppositeAdjacent

** Q How to remember trigonometry formulas?** We have many formulas in the higher classes that might be difficult to remember, so there are few steps to follow for remembering them:

Ans:

1. Get familiar with mathematical symbols.

2. Then comes the structure of the formulas and how they are derived.

3. Practice the formulas regularly.

4. Use flashcards of the formulas, then revise and finally test yourself.

** Q What is the basic formula of trigonometry?**We have six fundamental trigonometric ratios used in Trigonometry. These ratios are also known as trigonometric functions. 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 the Sine, cosine, tangent, Secant, cosecant, and cotangent values using trigonometric formulas.

Ans:

1. sinθ=OppositeSideHypotenusesinθ=OppositeSideHypotenuse

2. cosθ=AdjacentSideHypotenusecosθ=AdjacentSideHypotenuse

3. tanθ=OppositeSideAdjacentSidetanθ=OppositeSideAdjacentSide

4. secθ=HypotenuseAdjacentSidesecθ=HypotenuseAdjacentSide

5. θ=HypotenuseOppositeSideθ=HypotenuseOppositeSide

6. cotθ=AdjacentSideOppositeSidecotθ=AdjacentSideOppositeSide

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Cosine Function: ( cos (theta ) = frac{{{rm{ Adjacent }}}}{{{rm{ Hypotenuse }}}})

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1. ( sin theta = frac{{{rm{ Opposite, Side }}}}{{{rm{ Hypotenuse }}}})

2. ( cos theta = frac{{{rm{ Adjacent, Side }}}}{{{rm{ Hypotenuse }}}})

3. ( tan theta = frac{{{rm{ Opposite, Side }}}}{{{rm{ Adjacent, Side }}}})

4. ( sec theta = frac{{{rm{ Hypotenuse }}}}{{{rm{ Adjacent, Side }}}})

5. (theta = frac{{{rm{ Hypotenuse }}}}{{{rm{ Opposite, Side }}}})

6. ( cot theta = frac{{{rm{ Adjacent, Side }}}}{{{rm{ Opposite, Side }}}})"

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1. Basic formulas

2. Reciprocal Identities

3. Trigonometric Ratio Table

4. Periodic Identities

5. Co-function Identities

6. Sum and Difference Identities

7. Half-Angle Identities

8. Double Angle Identities

9. Triple Angle Identities

10. Product Identities

11. Sum of Product Identities"

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(theta + theta = 1)

(theta + 1 = theta )

(1 + theta = theta )"

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In the given article, we talked about all trigonometry formulas like basic formulas, reciprocal identities, trigonometric ratio table, periodic identities, cofunction identities, sum and difference identities, half-angle identities, double angle identities, and triple-angle product identities, and the sum of product identities. We discussed how to learn the formulas of trigonometry followed by the Inverse trigonometry formula. Then we discussed the trigonometry table formula and the trigonometry formulas list.

*Now we have covered all the Trigonometry formulas in this article. We hope you find them useful. Prepare for your exam using the same. If you think we have missed anything, or if you have suggestions, do let us know. We will be happy to hear from you, and update this article to add more value to it. Till then, stay tuned to Embibe for all updates on the trigonometry formulas for Class 10, exam preparation tips and the latest academic articles!*

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