The basic trigonometric functions include sine, cosine, and tangent, and they can be defined in terms of the ratios of the sides of a right triangle. 

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to that of the hypotenuse. 
• The cosine of an angle is the ratio of the length of the adjacent side to that of the hypotenuse. 
• The tangent of an angle is the ratio of the length of the opposite side to that of the adjacent side.

Introduction to Trigonometry Class 10 Notes

The introduction includes basic trigonometry concepts. These include the definition of trigonometric ratios and their representation in a right-angled triangle.

Trigonometric Ratios: Here, learn its definition and evaluation of trigonometric ratios of angles of any measure using the unit circle.

Trigonometric Ratios of Complementary Angles: It encompasses understanding the relationship between the trigonometric ratios of complementary angles.

Inverse Trigonometric Functions: Definition, range, domain and evaluation of inverse trigonometric functions are studied.

Trigonometric identities: Verifying trigonometric identities and solving trigonometric equations.

Applications of Trigonometry: These include solving problems involving heights and distances, angles of elevation and depression, and bearings in navigation.

Trigonometry of Circles: Understanding the trigonometry of circles, including the definition of cyclic quadrilaterals, arc length, area of a sector, and segments of a circle.

Heights and Distances: Solving problems involving heights and distances using trigonometry.

Trigonometric Ratios

It focuses on the ratios of the sides of a right triangle (a triangle with one angle equal to 90 degrees) in relation to its angles. These ratios are known as trigonometric ratios.

The six trigonometric ratios in a right triangle are:

 

Sine (sin) = opposite side/hypotenuse Cosine (cos) = adjacent side/hypotenuse Tangent (tan) = opposite side / adjacent side Cotangent (cot) = adjacent side / opposite side Secant (sec) = hypotenuse / adjacent side Cosecant (csc) = hypotenuse / opposite side

 

Here are some examples of using trigonometric ratios in a right triangle:

If the opposite side of an angle is 8 and the hypotenuse is 10, the sine of the angle is 8/10 or 0.8.

If the adjacent side of an angle is 6 and the hypotenuse is 10, the cosine of the angle is 6/10 or 0.6.

If the opposite side of an angle is 8 and the adjacent side is 6, the tangent of the angle is 8/6 or 1.33.

If the adjacent side of an angle is 6 and the opposite side is 8, the cotangent of the angle is 6/8 or 0.75.

If the hypotenuse of an angle is 10 and the adjacent side is 6, the secant of the angle is 10/6 or 1.67.

If the hypotenuse of an angle is 10 and the opposite side is 8, the cosecant of the angle is 10/8 or 1.25.

It’s important to remember that these ratios only apply to right triangles and that the side and angle conventions must be followed when using these ratios.

Some examples of trigonometric ratios in real-world scenarios:

In physics, trigonometry is used to calculate the force of gravity acting on an object based on its mass and distance from the centre of the Earth.

In engineering, trigonometry is used to determine the height and distance of tall structures like bridges and buildings.

In navigation, trigonometry is used to calculate the distance and bearing of a ship or aeroplane from a known point.

In cartography, trigonometry is used to make accurate maps and to calculate distances and angles on the Earth’s surface.

In astronomy, trigonometry is used to calculate the positions of celestial bodies and to map the universe.

It’s also used in fields such as computer graphics, signal processing, and many more.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables. They can be used to simplify trigonometric expressions and solve trigonometric equations. Here are some examples of trigonometric identities:

Pythagorean identities: These are based on the Pythagorean theorem and relate the sine, cosine, and tangent of an angle to itself and its supplementary angle. They are:

sin²x cos²x = 1

1 tan²x = sec²x

1 cot²x = csc²x

Reciprocal identities: These identities relate the reciprocal trigonometric functions to each other. They are:

cotx = 1/tanx

cscx = 1/sinx 

secx = 1/cosx

Quotient identities: These identities relate the quotient of two trigonometric functions to the product of the same functions. They are:

tanx = sinx/cosx

 cotx = cosx/sinx

cscx = 1/sinx

secx = 1/cosx

Co-function identities: These identities relate the sine and cosine functions to each other. They are:

cosx = sqrt(1-sin²x)

sin(90-x) = cosx

Even-odd identities: These identities relate the sine and cosine functions to each other. They are:

sin(-x) = -sin x 

cos(-x) = cos x

Periodic identities: These identities relate the period of trigonometric functions to pi. They are:

sin(x 2π) = sinx 

cos(x 2π) = cosx

It’s important to memorise these identities as they will be useful in solving trigonometric equations and in simplifying trigonometric expressions. It’s also important to practice applying these identities to solve problems.

Complementary Angles in Trigonometry Class 10 Notes

Complementary angles are two angles with combined measures adding up to 90 degrees.

Properties of Complementary Angles:

• The sum of the measures of complementary angles is always 90 degrees.
• The sine of one angle is equal to the cosine of the complement angle.
• The tangent of one angle is equal to the cotangent of the complement angle.
• The cosecant of one angle is equal to the secant of the complement angle.
• The product of the sine of one angle with the cosecant of the other angle is equal to 1.
• The product of the cosine of one angle with the secant of the other angle is equal to 1.

Examples of Complementary Angles:

• A 45-degree angle and a 45-degree angle.
• A 30-degree angle and a 60-degree angle.
• A 20-degree angle and a 70-degree angle.

It is important to note that complementary angles do not have to be acute angles (less than 90 degrees). They can also be obtuse angles (greater than 90 degrees) as long as the sum of the measures is 90 degrees.

In solving problems, the complementary angles are used to find the values of Trigonometric functions of one angle if we know the value of the other angle.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the six basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These inverse functions find the measure of an angle in a right triangle when the ratio of the sides is known.

The inverse sine function (arcsin) finds the measure of an angle in degrees or radians that has a sine value equal to a given number. The inverse cosine function (arccos) finds the measure of an angle that has a cosine value equal to a given number. The inverse tangent function (arctan) finds the measure of an angle that has a tangent value equal to a given number.

The inverse cotangent function (arccot) finds the measure of an angle that has a cotangent value equal to a given number. The inverse secant function (arcsec) finds the measure of an angle that has a secant value equal to a given number. The inverse cosecant function (arccsc) finds the measure of an angle that has a cosecant value equal to a given number.

It’s worth noting that for all trigonometric functions, their respective inverse functions will have a domain of (-1,1), and the range of the inverse function will be in the interval of (-90,90) in degree and (-π/2, π/2) in radians. Also, it’s important to remember that the inverse trigonometric functions give only the principal value of the angle.

Trigonometric Equations

Trigonometric equations involve trigonometric functions, such as sine, cosine, and tangent, and are used to find the values of an angle in a right triangle.

There are different types of trigonometric equations, each with its own method of solving:

Simple Trigonometric Equations: These equations involve only one trigonometric function and a constant. For example, sin x = 0.5, where x is the angle whose value we are trying to find.

Multiple-Angle Trigonometric Equations: These equations involve more than one angle and trigonometric functions. For example, sin 2x = cos x, where x is the angle whose value we are trying to find.

Trigonometric Equations With Quadratic Equations: These equations involve trigonometric functions and quadratic equations. For example, sin^2 x cos^2 x = 1, where x is the angle whose value we are trying to find.

Trigonometric Equations With Higher Degrees: These equations involve trigonometric functions and higher-degree polynomials.

It’s important to remember that for each trigonometric equation, we have to check the solution in the domain of the trigonometric function that we are using. Also, it’s important to remember that the solution of the trigonometric equations will not be a single value but a set of values.

Heights And Distances In Trigonometry Class 10 Notes

Heights and distances in trigonometry involve finding the height or distance of an object or structure. It refers to the application of trigonometry to real-world objects, such as a building or a tree. It is based on the measure of angles and the lengths of sides in a right triangle.

One of the most common methods used to solve these problems is the technique of triangulation. Triangulation involves measuring the angle of elevation or the angle of depression and the distance from the observer to a point on the ground directly beneath the object.

•  The angle between the horizontal line of sight and the line of sight to the top of an object is called the angle of elevation. 
• The angle between the horizontal line of sight and the line of sight to the bottom of an object is called the angle of depression.

Another method is to use the concept of similar triangles and trigonometric ratios. For example, if two similar triangles share an angle, we can use the ratio of corresponding sides to find the height of an object or the distance of a point.

It’s important to note that trigonometry is used to solve this type of problem, but it requires the use of basic algebraic manipulation, geometry, and trigonometry formulas and identities.

Some Important Trigonometry Formulas 

• sin^2 x cos^2 x = 1
• 1 tan^2 x = sec^2 x
• 1 cot^2 x = csc^2 x
• sin(x y) = sin x cos y cos x sin y
• cos(x y) = cos x cos y – sin x sin 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) = (tan x tan y) / (1 – tan x tan y)
• cot(x y) = (cot x cot y – 1) / (cot x cot y)

Inverse Trigonometry Functions

• arcsin(x) = sin^-1(x)
• arccos(x) = cos^-1(x)
• arctan(x) = tan^-1(x)
• arccot(x) = cot^-1(x)
• arcsec(x) = sec^-1(x)
• arccsc(x) = csc^-1(x)

Real-Life Application Of Trigonometry Class 10 Notes

Trigonometry has many real-life applications in fields such as engineering, physics, architecture, and navigation. Some examples of these applications include the following:

Navigation: Trigonometry is used to determine the position of ships and aeroplanes using triangulation. The position of the ship or aeroplane can be determined by measuring the angles of elevation or depression and the distances from a ship or aeroplane to three known points on the ground.

Surveying: Trigonometry is used in surveying to determine the distance, elevation, and angles of land features such as hills and valleys. The height of an object or the distance to a point can be determined by measuring the angles of elevation or depression and the distance to a point on the ground.

Architecture: Trigonometry is used to design and construct buildings and other structures. Architects use trigonometry to determine the height and slope of roofs, the incline of stairs, and the angles of beams and other structural elements.

Engineering: Trigonometry is used in engineering to design and analyse structures such as bridges and towers. Engineers use trigonometry to determine the forces and stress acting on the structures and to ensure that they are safe and stable