Hi friends, today we are going to study about Similarity Properties, Transformations, Trigonometric Ratios, and Pythagorean Triples for Grade XI. Firstly the introduction about trigonometry. Trigonometry deals with triangles, the right-angled triangle, circles, oscillations, and waves. When two right angle triangles are together then the result is supplementary composite triangles.
If one angle of a triangle is 90 degrees and other one is known, the third is fixed, because the sum of three angles of a triangle is 180 degrees. If the angles are known, the ratios of the sides are determined and if the length of one of the sides is known, the other two are determined. Let a, b and c is the length of triangle sides and A is known angle then ratio of trigonometric functions are,
1. Sine function is defined as the ratio of the side opposite the angle to the hypotenuse. (see for more information)
Sin A= opposite/ hypotenuse=a/c
2. Cosine functions are defined as the ratio of the adjacent leg to the hypotenuse.
Cos A= adjacent/ hypotenuse=b/c
3. Tangent functions are defined as the ratio of the opposite leg to the adjacent leg.
Tan A = opposite/ adjacent = a/b = Sin A/Cos A
4. Cosecant (A) is the reciprocal of sin (A). The ratio of the length of the hypotenuse to the length of the opposite side is called Cosecant.
Csc A = 1/sin A = hypotenuse/opposite = c/a
5. Secant (A) is the reciprocal of cos (A) and it is the ratio of the length of the hypotenuse to the length of the adjacent side.
Sec A = 1/cos A = hypotenuse/adjacent = b/c
6. Cotangent (A) is the reciprocal of tan(A) and defined as the ratio of the length of the adjacent side to the length of the opposite side.
Cot A = 1/tan = adjacent/ opposite=b/a=cos/sin.
The values of angles of these trigonometric functions can be easily computed by using the Pythagorean Theorem (Do you know What is Pythagorean Theorem). Similarly, the values of sine, cosine and tangent of an angle of π / 4 radians (45°) can also be found using the Pythagorean Theorem. Some values of angles are given below,
Sin π/4=sin450=cos π/4=cos450=1/√2,
Tan π/4 = tan 450 = 1.
Sin π/4=sin300=cos π/3=cos 600=1/2.
cos π/6=cos300=sin π/3=sin600= Ñ´3/2
tan π/6=tan300=cot π/3=cot600=1/Ñ´3.
Below table shows the values of different angles of different functions.
Now I am going to tell you the properties of trigonometry. The trigonometry properties are based on the law of sin, law of cos and law of tan.
Law of sin is a/sin A = b/sin B = c/sin c = 2R. Sine law can be used to calculate the area of a triangle if two angles and one side are known. Area = 1/2( ab sin C).
Law of cos: c2=a2+b2-2ab cos C. This law used to determine a side of a triangle if two sides and the angle between them are known.
Law of tan: a-b / a+b =tan[1/2(A-B)]/tan[1/2(A+B)].
From the above discussion I hope that it would help you to understand the properties and transformations, trigonometric ratios.
In upcoming posts we will discuss about Coordinate Geometry in Grade XI and Rotations. Visit our website for information on Central Board of Secondary Education
If one angle of a triangle is 90 degrees and other one is known, the third is fixed, because the sum of three angles of a triangle is 180 degrees. If the angles are known, the ratios of the sides are determined and if the length of one of the sides is known, the other two are determined. Let a, b and c is the length of triangle sides and A is known angle then ratio of trigonometric functions are,
1. Sine function is defined as the ratio of the side opposite the angle to the hypotenuse. (see for more information)
Sin A= opposite/ hypotenuse=a/c
2. Cosine functions are defined as the ratio of the adjacent leg to the hypotenuse.
Cos A= adjacent/ hypotenuse=b/c
3. Tangent functions are defined as the ratio of the opposite leg to the adjacent leg.
Tan A = opposite/ adjacent = a/b = Sin A/Cos A
4. Cosecant (A) is the reciprocal of sin (A). The ratio of the length of the hypotenuse to the length of the opposite side is called Cosecant.
Csc A = 1/sin A = hypotenuse/opposite = c/a
5. Secant (A) is the reciprocal of cos (A) and it is the ratio of the length of the hypotenuse to the length of the adjacent side.
Sec A = 1/cos A = hypotenuse/adjacent = b/c
6. Cotangent (A) is the reciprocal of tan(A) and defined as the ratio of the length of the adjacent side to the length of the opposite side.
Cot A = 1/tan = adjacent/ opposite=b/a=cos/sin.
The values of angles of these trigonometric functions can be easily computed by using the Pythagorean Theorem (Do you know What is Pythagorean Theorem). Similarly, the values of sine, cosine and tangent of an angle of π / 4 radians (45°) can also be found using the Pythagorean Theorem. Some values of angles are given below,
Sin π/4=sin450=cos π/4=cos450=1/√2,
Tan π/4 = tan 450 = 1.
Sin π/4=sin300=cos π/3=cos 600=1/2.
cos π/6=cos300=sin π/3=sin600= Ñ´3/2
tan π/6=tan300=cot π/3=cot600=1/Ñ´3.
Below table shows the values of different angles of different functions.
| Function | 00 | 300 | 450 | 600 | 900 | ||
| sin | 0 | 1/2 | Ñ´2/2 | Ñ´3/2 | 2 | ||
| Cos | 1 | Ñ´3/2 | Ñ´2/2 | 1/2 | 0 | ||
| Tan | 0 | Ñ´3/3 | 1 | Ñ´3 | infinity | ||
| Cot | infinity | Ñ´3 | 1 | Ñ´3/3 | 0 | ||
| Sec | 1 | 2Ñ´3/3 | Ñ´2 | 2 | infinity | ||
| Csc | infinity | 2 | Ñ´2 | 2 Ñ´3/3 | 0 |
Now I am going to tell you the properties of trigonometry. The trigonometry properties are based on the law of sin, law of cos and law of tan.
Law of sin is a/sin A = b/sin B = c/sin c = 2R. Sine law can be used to calculate the area of a triangle if two angles and one side are known. Area = 1/2( ab sin C).
Law of cos: c2=a2+b2-2ab cos C. This law used to determine a side of a triangle if two sides and the angle between them are known.
Law of tan: a-b / a+b =tan[1/2(A-B)]/tan[1/2(A+B)].
From the above discussion I hope that it would help you to understand the properties and transformations, trigonometric ratios.
In upcoming posts we will discuss about Coordinate Geometry in Grade XI and Rotations. Visit our website for information on Central Board of Secondary Education
No comments:
Post a Comment