Trigonometry Formulas for Class 10

Trigonometry formulas for Class 10 deal with the connections between the sides and angles of a right-angled triangle. These formulas, known as trigonometric ratios, express the relationships between the different sides of the triangle. All trigonometry formulas class 10 play a crucial role in determining angles and side lengths using these ratios. Beyond mathematics, these formulas have widespread applications in fields such as architecture, engineering, astronomy, geography, music, electronics, civil engineering, oceanography, image compression, chemistry, and medical imaging. Explore the significance and practical uses of these important class 10th trigonometry formulas.

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.

List of All Formulas of Trigonometry Class 10 

Discover trigonometry all formulas Class 10 in one handy list. This resource simplifies complex math concepts, making it easy for students to grasp essential geometry equations. Streamline your study routine and ace your Class 10 exams with this comprehensive guide.

Trigonometric Ratios:

The Three Main Trigonometric Ratios are:

The Resiprocal of the Above Ratios are:

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:

10th Trigonometric Formulas:

Trigonometric formulas are like secret codes that help us solve triangles and angles in math. They're simple rules that make it easy to find missing pieces in geometric puzzles. Whether it's calculating angles or sides, these formulas are your math superpowers!

All formulas of Trigonometric Class 10 and identities are based on the right-angled triangle. They are:

1. Pythagorean Formula:

In the 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 in Trigonometry Formulas for Class 10:

The concept of congruence in trigonometry doesn't exist in the same way it does in geometry. Trigonometry deals with relationships between angles and sides in triangles, not the shapes themselves. However, there are some principles that are relevant to congruency trigonometry all formula class 10:

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 in Class 10th Trigonometry Formula:

Trigonometry and theorems on similarity in geometry work hand-in-hand, but trigonometry itself doesn't have specific theorems dedicated to similarity. However, the concept of similar triangles plays a crucial role in solving trigonometric problems. Here are some key points to remember in maths formula for class 10 trigonometry:

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, 

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

Applications of all Formulas of Trigonometry Class 10

Trigonometry all formula class 10, finds wide applications in different fields such as construction, engineering, design, seismology, oceanography, phonetics, and architecture. These formulas are used for various calculations in these areas. Let's explore some practical uses:

1. Height and Distance: formulas for Trigonometry class 10 help in determining the height of buildings, mountains, towers, as well as calculating angles of elevations and slopes.

2. Navigation: In real-life scenarios, maths formula for class 10 trigonometry comes in handy for calculating the distance between a ship and the seashore.

3. Astronomy: Astronauts rely on trigonometry formulas to figure out the speed and altitude of celestial bodies.

4. Physics Applications: All Trigonometric formulas Class 10 play a crucial role in representing physical quantities like oscillations, electromagnetic waves, and mechanical waves.

5. Solar Panels: In the realm of sustainable energy, trigonometry is applied in solar cell panels to calculate the angle of the sun's height. This information is crucial for determining power output at different angles.

In essence, All formulas of trigonometry class 10 have practical implications in our everyday lives, contributing to a variety of fields and helping us solve real-world problems. Download trigonometry formulas for class 10 PDF now!

Tips to Memorize All Formulas of Trigonometry Class 10

Here are some straightforward tips for remembering all trigonometry formulas Class 10:

1. Get a good grasp of the terms related to ratios and angles. This foundation is crucial for applying trigonometry formulas effectively.

2. Address any confusion by contacting our Vedantu’s Doubt expert-led study materials or reaching out to your teacher. Clarifying uncertainties early on is key to solidifying your understanding.

3. Enhance your understanding by going through the step-by-step derivation of trigonometry formulas for class 10. This approach builds a strong foundation for grasping the concepts involved.

4. Make memorization fun by using phrases or mnemonic devices. This can make it easier to recall the class 10th trigonometry formulas when you need them.

5. Download trigonometry formulas PDF class 10, this handy copy can help you quickly revise the formulas whenever you want.

6. Reinforce your learning by solving numerous problems and examples from the textbook. This will give you a comprehensive understanding of how to use the formulas in different situations.

Conclusion

Mastering the trigonometry formulas for class 10 is essential for success in mathematics. Vedantu offers a comprehensive guide, including a trigonometry formula for class 10 PDF download, covering all the vital formulas. From sine, cosine, and tangent to the Pythagorean theorem, these formulas play a crucial role in solving trigonometric problems at the 10th-grade level. Whether it's finding angles, sides, or understanding the relationships between them, these formulas serve as the foundation. Students can access the all formulas of trigonometry class 10 on Vedantu, making learning accessible and effective. Embracing these formulas empowers students to tackle challenges with confidence and excel in their mathematical journey.