All trigonometric formulas for Class 10 PDF

What is Trigonometry?

The word Trigonometry is derived from the Greek word ‘Trigonon’ which means ‘triangle’ and ‘metron’ which refers to the term ‘measure’. It is the 16th century Latin derivative. This concept of Trigonometry was given by Greek Mathematician Hipparchus. According to Victor Katz in “A History of Mathematics (3rd edition)”, Trigonometry was developed primarily from the needs of Greek and Indian Astronomers.

Trigonometry is the most important concept in Mathematics. It deals with the sides and angles of the right-angled triangle. It plays the most vital role in almost all fields , whether it’s aviation, physics, criminology, military, marine biology, development of sound waves, satellite navigation, medical imaging etc. Trigonometry is used for finding the angles or the sides of the right-angled triangle.

Right-Angled Triangle:

Here, in the figure, a right-angled triangle is shown having hypotenuse (the longest side), base (adjacent side), height (opposite side), and angle Ө.    

This triangle is of great importance because if anyone tries to find the direct distance and angle, then that can be easily found using this.

The basic functions of Trigonometry are sine, cosine, and tangent. The other three functions cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively.

Trigonometric Ratios:

The Three Main Trigonometric Ratios are:

\[sin\theta= \frac{\textrm{Opposite Side}}{\textrm{Hypotenuse Side}}\]

\[cos\theta= \frac{\textrm{Adjacent Side}}{\textrm{Hypotenuse Side}}\]

\[tan\theta= \frac{\textrm{Opposite Side}}{\textrm{Adjacent Side}}\]

The Inverse of the Above Ratios are:

\[sec\theta=\frac{1}{cos\theta }= \frac{\textrm{Hypotenuse Side}}{\textrm{Adjacent Side}}\]

\[cosec\theta=\frac{1}{sin\theta }= \frac{\textrm{Hypotenuse Side}}{\textrm{Opposite Side}}\]

\[cot\theta=\frac{1}{tan\theta }= \frac{\textrm{Adjacent Side}}{\textrm{Opposite Side}}\]

Below are the Relation between the Trigonometric Identities:

\[tan\theta =\frac{sin\theta }{cos\theta }\]

\[cot\theta =\frac{cos\theta }{sin\theta }\]

Trigonometric Angles:

In Trigonometry, there are five angles. Other angles can also be found but these are the basics. These angles are 00, 300, 450, 600, 900 . The table for the same is given below:

Trigonometric Formulas:

Trigonometric identities and formulae are based on the right-angled triangle. They are:

1. Pythagorean Formula:

For the above right-angled triangle, the sum of the squares of base and height is equal to the square of the hypotenuse.

Thus, 

\[a^2+b^2=c^2\]

And, according to the Pythagoras Theorem,

• \[sin^2\theta +cos^2\theta =1\]

• \[tan^2\theta +1=sec^2\theta \]

• \[cot^2\theta +1=cosec^2\theta \]

• \[sin2\theta =2sin\theta cos\theta \]

• \[cos2\theta =cos^2\theta -sin^2\theta \]

• \[tan2\theta =\frac{2tan\theta}{1-tan^2\theta }\]

• \[cot2\theta =\frac{cot^2\theta -1}{2cot\theta }\]

2. Sum and Difference Identities:

For two angles u and v, identities related to sum and difference of these two angles are as below:

• \[sin(u+v)=sin(u)cos(v)+cos(u)sin(v)\]

• \[cos(u+v)=cos(u)cos(v)-sin(u)sin(v)\]

• \[tan(u+v)=\frac{tan(u)+tan(v)}{1-tan(u)tan(v)}\]

• \[sin(u-v)=sin(u)cos(v)-cos(u)sin(v)\]

• \[cos(u-v)=cos(u)cos(v)+sin(u)sin(v)\]

• \[tan(u-v)=\frac{tan(u)-tan(v)}{1+tan(u)tan(v)}\]

3. Reduction Formulas:

The angles of any other quadrants can be reduced to the equivalent first quadrant angle. This can be done by changing the signs and Trigonometric ratios. The reduction formulas for the same are:

First Quadrant

• \[sin(90-\theta)=cos\theta \]

• \[cos(90-\theta)=sin\theta \]

• \[tan(90-\theta)=cot\theta \]

• \[csc(90-\theta)=sec\theta \]

• \[sec(90-\theta)=csc\theta \]

• \[cot(90-\theta)=tan\theta \]

Second Quadrant

• \[sin(180-\theta)=sin\theta \] 

• \[cos(180-\theta)=-cos\theta \] 

• \[tan(180-\theta)=-tan\theta \]

• \[csc(180-\theta)=csc\theta \]  

• \[sec(180-\theta)=-sec\theta \] 

• \[cot(180-\theta)=-cot\theta \] 

Third Quadrant

• \[sin(180+\theta)=-sin\theta \] 

• \[cos(180+\theta)=-cos\theta \]

• \[tan(180+\theta)=tan\theta \] 

• \[csc(180+\theta)=-csc\theta \] 

• \[sec(180+\theta)=-sec\theta \] 

• \[cot(180+\theta)=cot\theta \] 

Fourth Quadrant

• \[sin(360-\theta)=-sin\theta \]

• \[cos(360-\theta)=cos\theta \]

• \[tan(360-\theta)=-tan\theta \]

• \[csc(360-\theta)=-csc\theta \]

• \[sec(360-\theta)=sec\theta \]

• \[cot(360-\theta)=-cot\theta \]   

Congruent Triangles:

Two triangles are congruent if they are superimposed on each other. The term “Congruent” defines the object and its mirror image. 

The two triangles must be congruent if they have the same length of sides and the same measure of angles. Thus, they can be superimposed on each other. Congruence can be represented by the symbol.

\[\cong\] 

Rules for Congruency:

1. SSS(Side-Side-Side)

If the two triangles have the equivalent corresponding sides, then these two triangles will be congruent by the SSS rule.

For example,

\[\textrm{AC=PR}\]

\[\textrm{BC=QR}\]

\[\textrm{AB=PQ}\]

In the above two triangles ABC and PQR, Images are to be uploaded soon.

Here, the triangles ABC and PQR are congruent by the SSS rule because the corresponding sides of these two triangles are equivalent.

Thus,

\[\Delta \textrm{ABC}\cong \Delta \textrm{PQR}\]

2. SAS (Side-Angle-Side)

If the two triangles have equivalent two corresponding sides and also the angles made up by these corresponding sides are equivalent, then these triangles will be congruent by SAS rule.

For example,

In the above two triangles ABC and PQR, Images are to be uploaded soon.

Here, the triangles ABC and PQR are congruent by the SAS rule because the corresponding two sides and the angles made up by these sides are equivalent.

Thus,

3. ASA (Angle-Side-Angle)

If the two triangles have equivalent two corresponding angles and also, the sides between these corresponding angles are equivalent, then these triangles will be congruent by the ASA rule.

For example,

In the above two triangles ABC and PQR, Images are to be uploaded soon.

\[\textrm{AB=PQ}\]

\[\textrm{AC=PR}\]

\[\angle A=\angle P\]

Here, the triangles ABC and PQR are congruent by the ASA rule because the corresponding two angles and the sides between these angles are equivalent.

Thus,

\[\Delta ABC\cong \Delta PQR\]

4. RHS (Right Angle-Hypotenuse-Side)

If the hypotenuses and the corresponding sides of the two right-angled triangles are equivalent, then these two right-angled triangles will be congruent by the RHS rule.

For example,

In the above two triangles XYZ and RST, Images are to be uploaded soon.

\[\textrm{XZ=RT}\]

\[\textrm{YZ=ST}\]

Here, the triangles XYZ and RST are congruent by the RHS rule because the hypotenusesXZ and RT and the corresponding sides YZ and ST of the right-angled triangles are equivalent.

Thus, 

\[\Delta XYZ\cong \Delta RST\]

Similar Triangles

Two triangles will be similar if they have the same angles and different lengths of sides. The similarity of the two triangles is represented by the symbol ~. 

Thus, the two triangles must be similar if they have equal corresponding angles and the sides are in proportion.

For example,

In the above two triangles ABC and XYZ, Images are to be uploaded soon.

\[\angle A=\angle X,\angle B=\angle Y\textrm{and} \angle C=\angle Z\]

\[\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}\]

Rules for Similarity:

1. AAA (Angle-Angle-Angle)

Two triangles will be similar by the AAA rule if they have equal corresponding angles. For example,

In the above two triangles ABC and DEF,

\[\angle A=\angle D,\angle B=\angle E \textrm{and} \angle C=\angle F\]

Here, the triangles ABC and DEF are similar by the AAA rule because the corresponding angles of these two triangles are equal.

Thus,

\[\Delta ABC\sim \Delta DEF\]

2. SSS (Side-Side-Side)

Two triangles will be similar by the SSS rule if the corresponding sides of the triangles are in proportion. For example,

In the above two triangles ABC and DEF, Images are to be uploaded soon.

Here, the triangles ABC and DEF are similar by the SSS rule because the corresponding sides of these two triangles are in proportion.

3. SAS (Side-Angle-Side)

The two triangles will be similar by the SAS rule if the two corresponding sides are in proportion and the angles between these corresponding sides are equal.

For example,

In the above two triangles LMN and QRS, Images are to be uploaded soon.

\[\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\]

\[\frac{12}{6}=\frac{16}{8}=\frac{18}{9}\]

\[2=2=2\]

Here, the triangles LMN and QRS are similar by SAS rule because the two corresponding sides of these two triangles are in proportion and the angles between these two corresponding sides are equal.

Thus, 

\[\Delta ABC\sim \Delta DEF\]

Theorems on Similarity

If the two triangles are similar, then the ratio of their areas must be in proportion with the squares of the ratio of their sides.

For similar triangles ABC and DEF, Images are to be uploaded soon.

\[\frac{\textrm{Area of ABC}}{\textrm{Area of DEF}}=\left ( \frac{AB}{DE} \right )^2=\left ( \frac{BC}{EF} \right )^2=\left ( \frac{CA}{FD} \right )^2\]

Law of Sine: 

If A, B, and C are angled and a, b, and c are the sides of a triangle, then:

(Image will be uploaded soon)

How many trigonometric formulas are there in class 10?

What are the 45 formulas of trigonometry?

What are the 9 identities of trigonometry?

How many total formulas are there in trigonometry?