Trigonometry Table

Trigonometry is among the most essential branches of mathematics that has a wide range of applications. Trigonometry is the discipline of mathematics that studies the relationship between the lengths and angles of a right-angle triangle. 

As a result, using trigonometric formulas, functions, or identities, it is possible to find the missing or unknown angles or sides of a right triangle. 0°, 30°, 45°, 60°, and 90° are some of the most regularly utilized trigonometric angles in calculations.

Trigonometry is divided further into two branches. The following are the two types of trigonometry:

• Plane Trigonometry

• Spherical Trigonometry

Trigonometry Table:

The values of trigonometric standard angles like 0°, 30°, 45°, 60°, and 90° can be found using the trigonometric ratios table. The trigonometric ratios are Sine, cosine, tangent, cosecant, secant, and cotangent. Sin, cos, tan, cosec, sec, and cot are abbreviated forms of these ratios. 

To answer trigonometry issues, the value of trigonometric ratios of standard angles are required. As a result, the value of the trigonometric ratios of such standard angles must be remembered.

Refer to the table for typical angles that can be utilized to solve numerous trigonometric ratio problems.

Similarly, we may determine the trigonometric ratio values for angles other than 90° , such as 180°, 270°, and 360°.

Unit Circle:

We can use trigonometric functions to every angle, even ones greater than 90°, by using the unit circle. The periodicity of trigonometric functions is demonstrated by the unit circle, which shows that they produce a set of numbers at regular intervals.

Because the center of the circle is at the origin and the radius is 1, the idea of a unit circle allows us to accurately measure the angles of cos, sin, and tan. Assume that theta is an angle.

Assume that the length of the perpendicular is y and that the length of the base is x. The hypotenuse is the same length as the radius of the unit circle which is also 1. As a result, the trigonometric ratios can be written as;

Trigonometric Identities:

The identities are geometrically related to specific trigonometric functions (like sine, cosine, and tangent) of one or even more angles.

The major trigonometry functions are sine, cosine, and tangent, while the other 3 functions are cotangent, secant, and cosecant. 

Only the right-angle triangle has the trigonometric identities. The sides of the right triangle, like the adjacent, opposing, and hypotenuse sides, are used to define all of these trigonometric ratios. 

Trigonometric Identities List:

In trigonometry, numerous identities are utilized to solve a variety of trigonometric problems. Complex trigonometric problems can be addressed rapidly using these trigonometric identities or formulas. Let’s look at all of the basic trigonometric identities.

Reciprocal Identities in Trigonometry

The following are the reciprocal trigonometric identities:

• Sin ϴ = 1/cosecϴ 

• cosϴ = 1/secϴ

• tanϴ = 1/cotϴ 

Trigonometric Pythagorean Identities

In trigonometry, there are 3 Pythagorean trigonometric identities based on right-triangle theorem also called Pythagoras theorem.

• sin2 a + cos2 a = 1

• 1+tan2 a = sec2 a

• cosec2 a = 1 + cot2 a

Identities of Opposing Angles in Trigonometry

The following are examples of opposing angle trigonometric identities:

• Sin (-ϴ) = – Sin ϴ

• Cos (-ϴ) = Cos ϴ

• Tan (-ϴ) = – Tan ϴ

• Cot (-ϴ) = – Cot ϴ

• Sec (-ϴ) = Sec ϴ

Complementary Angles’ Trigonometric Identities

Two angles are said to be complementary in geometry if their sum equals 90°. Similarly, the trigonometric identities for complementary angles can be learned here.

• Sin (90° – ϴ) = Cos ϴ

• Cos (90° – ϴ) = Sin ϴ

• Tan (90° – ϴ) = Cot ϴ

• Cot (90° – ϴ) = Tan ϴ

• Sec (90° – ϴ) = cosec ϴ

• Cosec (90° – ϴ) = Sec ϴ

Supplementary Angles’ Trigonometric Identities

If the sum of two angles equals 90 degrees, they are supplementary. Similarly, the trigonometric identities for supplementary angles can be learned here.

• sinϴ = sin (180°- θ) 

• -cos θ = cos (180°- θ) 

• cosec θ = cosec (180°- θ) 

• -sec θ = sec (180°- θ)

• -tan θ = tan (180°- θ) 

• -cot θ = cot (180°- θ) 

Identities with a Double Angle

The trigonometry identities for sin, cos, and tan when the angles are doubled are:

• 2 sinϴ cosϴ = sin 2ϴ

• 2 cos2ϴ – 1 = 1 – 2sin2 ϴ = cos 2ϴ = cos2ϴ – sin2 ϴ 

•  (2tanϴ)/ (1 – tan2ϴ) = tan 2ϴ 

Product-Sum Identities in Trigonometry

The sum and difference of sines or cosines is transformed into a product of sines and cosines using the product-sum trigonometric identities.

• Sin C + Sin D = 2 Sin(C+D)/2. Cos(C-D)/2

• Cos C + Cos D = 2 Cos(C+D)/2. Cos(C-D)/2

• Sin C – Sin D = 2 Cos(C+D)/2. Sin(C-D)/2

Conclusions:

Trigonometry can be used to roof a house, make the roof inclined (in the case of single-family bungalows), and calculate the height of a building’s roof, among other things. The navy and aviation industries use it. Cartography makes use of it (creation of maps). 

In physics, trigonometry is used to find the components of vectors, model the mechanics of waves & oscillations add field strengths, and apply dot and cross products. Trigonometry has several applications in projectile motion.