diploma trigonometry
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PLEASE REFER TO PG NO 32 of 10th class geometry book...
IN SURE THIS WILL HELP YOU...
Answer:Trigonometry is one of the important branches in the history of mathematics and this concept is given by a Greek mathematician Hipparchus. Basically, it is the study of triangles where we deal with the angles and sides of the triangle. To be more specific, its all about a right-angled triangle. It is one of those divisions in mathematics that helps in finding the angles and missing sides of a triangle by the help of trigonometric ratios. The angles are either measured in radians or degrees. The usual trigonometry angles are 0°, 30°, 45°, 60° and 90°, which are commonly used.
Trigonometry Table
Trigonometry For Class 10
Trigonometry For Class 11
Trigonometry Formulas for Class 10
Trigonometry Formulas for Class 11
Trigonometry Formulas for Class 12
This branch divides into two sub-branches called plane trigonometry and spherical geometry. Here in this theory, you will learn about the trigonometric formulas, functions, and ratios, Right-Angled Triangles, etc.
Trigonometry Ratios-Sine, Cosine, Tangent
The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos, and tan. Let us see how are these ratios or functions, evaluated in the case of a right-angled triangle.
Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse is referred to as the adjacent and opposite.
Trigonometry Ratios
Six Important Trigonometric Functions
The six important trigonometric functions (trigonometric ratios) are calculated by the below formulas using above figure. It is necessary to get knowledge regarding the sides of the right triangle, because it defines the set of important trigonometric functions.
Functions Abbreviation Relationship to sides of a right triangle
Sine Function sin Opposite side/ Hypotenuse
Tangent Function tan Opposite side / Adjacent side
Cosine Function cos Adjacent side / Hypotenuse
Cosecant Function cosec Hypotenuse / Opposite side
Secant Function sec Hypotenuse / Adjacent side
Cotangent Function cot Adjacent side / Opposite side
Trigonometry Angles
The trigonometry angles which are commonly used in trigonometry problems are 0°, 30°, 45°, 60° and 90°. The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize. We will also show the table where all the ratios and their respective angle’s values are mentioned. To find these angles we have to draw a right-angled triangle, in which one of the acute angles will be the corresponding trigonometry angle. These angles will be defined with respect to the ratio associated with it.
For example, in a right-angled triangle,
Sin θ = Perpendicular/Hypotenuse
or θ = sin-1 (P/H)
Similarly,
θ = cos-1 (Base/Hypotenuse)
θ = tan-1 (Perpendicular/Base)
Trigonometry
Trigonometry Table
Check the table for common angles which are used to solve many trigonometric problems based on trigonometric ratios.
Angles 0° 30° 45° 60° 90°
Sin θ 0 ½ 1/√2 √3/2 1
Cos θ 1 √3/2 1/√2 ½ 0
Tan θ 0 1/√3 1 √3 ∞
Cosec θ ∞ 2 √2 2/√3 1
Sec θ 1 2/√3 √2 2 ∞
Cot θ ∞ √3 1 1/√3 0
Unit Circle
The concept of unit circle helps us to measure the angles of cos, sin and tan directly since the centre of the circle is located at the origin and radius is 1. Consider theta be an angle then,
Trigonometry - Unit Circles
Suppose the length of the perpendicular is y and of base is x. The length of the hypotenuse is equal to the radius of the unit circle, which is 1. Therefore, we can write the trigonometry ratios as;
Sin θ y/1 = y
Cos θ x/1 = x
Tan θ y/x
Trigonometry Formula
The Trigonometric formulas or Identities are the equations which are true in the case of Right-Angled Triangles. Some of the special trigonometric identities are as given below –
Pythagorean Identities
sin ² θ + cos ² θ = 1
tan 2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos² θ – sin² θ
tan 2θ = 2 tan θ / (1 – tan² θ)
cot 2θ = (cot² θ – 1) / 2 cot θ
Sum and Difference identities-
For angles u and v, we have the following relationships:
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) = 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) = tan(u) − tan(v)1+tan(u) tan(v)
If A, B and C are angles and a, b and c ar