Trigonometric ratios and identities form the backbone of Trigonometry, a mathematical branch that deals with the relationships and properties of triangles and angles. Trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent are Basic principles employed to delineate the connections among the angles and segments of a right-angled triangle.
1.0Trigonometric Ratios
Six trigonometric ratios are defined as sine, cosine, tangent, and their reciprocal resp as cosecant, secant, and cotangent based on the ratio of any two sides in a right-angled triangle.
Here are the key Trigonometric Ratios:
Sine (sin): The sine of an angle within a right triangle is established As the ratio between the length of the side opposite to the angle and the length of the hypotenuse.
sin(θ)= Hypotenuse Opposite Side or sinθ=HP
Cosine (cos): The cosine of an angle within a right triangle is defined as the ratio between the length of the adjacent side of the angle and the hypotenuse length.
cos(θ)= Hypotenuse AdjacentSide or cosθ=HB
Tangent (tan): The tangent of an angle within a right triangle is defined as the ratio between the length of the opposite side of the angle and the length of the adjacent side.
tan(θ)= Adjacent Side Opposite Side or tanθ=BP
Cosecant (cosec): The cosecant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side opposite the angle.
cosecθ= Opposite side Hypotenuse or cosecθ=PH
Secant (sec): The secant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side adjacent to the angle.
sec(θ)= Adjacent Side Hypotenuse or secθ=BH
Cotangent (cot): The cotangent of an angle within a right triangle is defined as the ratio between the adjacent side and the length of the side opposite the angle.
cot(θ)= Opposite Side Adjacent Side or cotθ=PB
These ratios are useful in solving various problems involving angles and distances in real-world applications such as Physics, Engineering, and Astronomy.
2.0Trigonometric Ratios and Identities Formulas
Trigonometric identities constitute equations that hold true for all values of the involved variables. They are derived from the definitions of trigonometric functions and are used extensively in simplifying trigonometric expressions and solving trigonometric equations. Some important 8 basic trigonometric identities include:
Understanding and applying these trigonometric ratios and identities are essential skills for students and professionals in fields related to mathematics and its applications.
5.0Sample Question from Trigonometric Ratios and Identities
What are the Pythagorean identities?
Ans: The Pythagorean identities are a set of three identities derived from the Pythagorean theorem. They are:
sin2(θ) + cos2(θ) = 1
tan2(θ) + 1 = sec2(θ)
1 + cot2(θ) = cosec2(θ)
Table of Contents
1.0Trigonometric Ratios
2.0Trigonometric Ratios and Identities Formulas
2.1Pythagorean Identities
2.2Reciprocal Identities
2.3Opposite Angle Identities
2.4Complementary Angles Identities
2.5Supplementary Angles Identities
2.6Sum and Difference Identities
2.7Sum to Product Identities
2.8Double - Angle Identities:
2.9Triple Angle Identities
2.10Half angle Identities
3.0Trigonometric Table
4.0Questions on Trigonometric Ratios and Identities
5.0Sample Question from Trigonometric Ratios and Identities
Home
Trigonometric ratios are mathematical relationships that describe ratios of the sides of a right triangle. The main trigonometric ratios are sine, cosine, and tangent, which are defined as the ratios of certain sides of a right triangle with respect to a given acute angle.
The primary trigonometric ratios are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively. These ratios are defined based on the relationships between the sides of a right triangle and an acute angle.
Trigonometric ratios are used to solve problems involving angles and sides of triangles. By knowing the values of certain trigonometric ratios for a given angle, we can find missing side lengths or angle measures in a triangle.
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are derived from the definitions of trigonometric functions and are useful for simplifying expressions and solving equations involving trigonometric functions.
Some common trigonometric identities used in solving equations include the sum and difference identities, double-angle identities, and half-angle identities. These identities allow us to express trigonometric functions of sums, differences, double angles, and half angles in terms of simpler functions.
Trigonometric ratios and identities are extensively used in various fields such as physics, engineering, astronomy, and surveying. They are applied to solve problems involving angles, distances, and periodic phenomena, making them essential tools in practical situations.
8 basic trigonometric Identities are- Pythagorean Identities, Reciprocal Identities, Sum of Difference Identities, Sum of Product Identities, Double - Angle Identities, Triple - Angle Identities, Half - Angle Identities and Trigonometric Product Identities