CBSE 10
CBSE 10: Your Epic Journey to Trigonometry Begins (Trigonometry Table)
May 7, 2024
Aha
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English
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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is an important part of geometry and has many real-world applications. In this article, we will explore the basics of trigonometry.
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to find the lengths of sides and the measures of angles in triangles. Trigonometry is also used to study the properties of periodic functions.
Trigonometry ratios are used to relate the angles of a triangle to its sides. The three main trigonometry ratios are sine, cosine, and tangent. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
There are six important trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These functions are used to relate the angles of a triangle to its sides.
Trigonometric functions can also be classified as even or odd functions. An even function is a function that is symmetric with respect to the y-axis. An odd function is a function that is symmetric with respect to the origin.
An even function is symmetric with respect to the y-axis. This means that if you reflect the graph of an even function across the y-axis, you get the same graph. The cosine function is an even function, as cos (-θ) = cos θ.
An odd function is symmetric with respect to the origin. This means that if you reflect the graph of an odd function across the origin, you get the same graph. The sine and tangent functions are odd functions, as sin (-θ) = -sin θ and tan (-θ) = -tan θ.
Understanding the even and odd properties of trigonometric functions can be useful in simplifying complex trigonometric expressions and solving trigonometric equations.
Inverse trigonometric functions are used to find the angle measures given the values of the trigonometric ratios. The inverse trigonometric functions are also called arc functions or anti-trigonometric functions.
The six inverse trigonometric functions are:
The range of each inverse trigonometric function is restricted to a certain interval to ensure that the function is one-to-one and has an inverse.
The inverse trigonometric functions are often used to solve trigonometric equations and to find the inverse derivatives of trigonometric functions in calculus.
For example, if we know that sin θ = 0.5, we can use the arcsine function to find the value of θ:
sin-1 0.5 = 30° or π/6 radians.
Similarly, if we know that tan θ = 1, we can use the arctangent function to find the value of θ:
tan-1 1 = 45° or π/4 radians.
It is important to note that the inverse trigonometric functions are only defined for certain values of the input. For example, the arcsine function is only defined for input values between -1 and 1.
A trigonometry table is a table of the values of the trigonometric functions for different angles.
The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used to define the values of the trigonometric functions for all angles in radians.
The unit circle can be used to derive the values of the trigonometric functions for the angles 0°, 30°, 45°, 60°, and 90°. For example, at 0°, the point on the unit circle is (1,0), which means that the sine of 0° is 0 and the cosine of 0° is 1.
The trigonometric functions can be defined as the ratios of the sides of a right triangle in relation to one of the acute angles in the triangle. However, these functions can also be defined using the unit circle. For example, the sine of an angle θ can be defined as the y-coordinate of the point on the unit circle that corresponds to the angle θ, and the cosine of an angle θ can be defined as the x-coordinate of the point on the unit circle that corresponds to the angle θ.
The trigonometric functions can also be defined using the unit circle in terms of radians. For example, the sine of an angle θ in radians can be defined as the y-coordinate of the point on the unit circle that corresponds to the angle θ, and the cosine of an angle θ in radians can be defined as the x-coordinate of the point on the unit circle that corresponds to the angle θ.
The values of the trigonometric functions for the angles 0°, 30°, 45°, 60°, and 90° are summarized in the trigonometry table above. These values can be used to find the values of the trigonometric functions for other angles using the unit circle and trigonometric identities.
Trigonometric identities are mathematical equations that relate the trigonometric functions to each other. These identities can be used to simplify complex trigonometric expressions and to solve trigonometric equations. Some of the most important trigonometric identities are listed in the section above.
Euler’s formula is a mathematical equation that relates the trigonometric functions to the exponential function. It is often used to simplify complex trigonometric expressions.
This is Euler's formula:
eiθ = cos θ + i * sin θ
where e is the base of the natural logarithm, i is the imaginary unit (i2 = -1), θ is the angle in radians, cos θ is the cosine of θ, and sin θ is the sine of θ.
This formula relates the exponential function to the trigonometric functions and is often used to simplify complex trigonometric expressions. The formula can also be used to derive various trigonometric identities and to solve differential equations in physics and engineering.
Trigonometry examples are problems that involve the use of trigonometric functions to solve for unknown values. Example problems include finding the length of a side of a triangle, finding the measure of an angle in a triangle, and finding the value of a trigonometric function for a given angle.
Find the exact value of cos (π/3).
✅ Solution: Use the table above: cos (π/3) = 1 / 2.
Trigonometry has many real-world applications, including in architecture, engineering, physics, and navigation. Trigonometry is also used in the study of waves, sound, and light.
Trigonometry problems can be solved using a variety of techniques, including the use of trigonometric functions, the Pythagorean theorem, and the Law of Sines and Law of Cosines. Solutions to trigonometry problems often involve finding the lengths of sides and the measures of angles in triangles.
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
The six basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.
The primary function of trigonometry is to find the lengths of sides and the measures of angles in triangles.
Trigonometry was developed by Greek mathematicians, including Hipparchus and Ptolemy.
Trigonometry has many real-world applications, including in architecture, engineering, physics, and navigation.
In conclusion, trigonometry is an important branch of mathematics that has many real-world applications. It is used to find the lengths of sides and the measures of angles in triangles, as well as to study the properties of periodic functions. If you are interested in learning more about trigonometry, check out Aha AI, a website that offers comprehensive resources and practice problems.
