The two rays that have the same beginning point that forms the figure called as an angle. The same beginning point is called the vertex and those two rays are called the sides of the angle which is also called as legs. Consider the given figure,

If OA is the initial side of the angle and OB is the terminal side of an angle, then we can say it is an oriented angle. The orientation of an angle is indicated by an arrow symbol where it starts from initial side to the terminal side and the angle is represented as ∠AOB. We can also draw an arrow in the opposite direction, starting from the initial side of the angle OA. Both angles represent the same oriented angle. The angle ∠BOA is opposite to the angle ∠AOB which is differently oriented.

The two common measurements used for determining angles are degree and radians. The most familiar unit of measurement of an angle is degree. Consider, a circle is divided into 360 equal parts, where we can get the right angle with 90^{0}. Further each degree is subdivided into 60 minutes and each minute is again subdivided into 60 seconds. The symbols used for degrees, minutes and seconds are °, ‘ and ” respectively. Minutes and seconds are also stated as arcminutes and arcseconds.

Beyond practical geometry in mathematics, angles are also used to measure in radians. The circle with the angle of 1 radian determines an arc with the length of radius. Because the length of the full circle is 2πr. Alternatively we can say that the circle contains 2π radians. The subdivision of radians are also written in decimal forms.

## Conversion Between Radians and Degrees

Note that, when an angle is represented in radians, only mention the value, not the term “rad”.

Because \(2\pi =360^{\circ}\), following conversion formulas has to be applied.

r \(rad\rightarrow \left [ \frac{360.r}{2\pi } \right ]^{\circ}\) \(g^{\circ}\rightarrow \left [ \frac{2\pi .g}{360} \right ]\) rad

## Trigonometry Angles Formula

sine, cosines and tangents, cotangents of some angles are equal to the sine, cosines and tangents, cotangents of other angles. Some of the trigonometry angle formulas given as per the figure are :

1 . Supplementary angles ( = sum is π)

- Sin ( π – α ) = sin α
- Cos (π – α ) = – cos α
- Tan (π – α ) = – tan α
- Cot (π – α) = – cot α

2 . Anti-supplementary Angles (= difference is π)

- Sin ( π + α ) = – sin α
- Cos (π + α ) = – cos α
- Tan (π + α ) = tan α
- Cot (π + α ) = cot α

3 . Opposite Angles ( = sum is 2π)

- Sin ( 2π – α ) = – sin α
- Cos (2π – α ) = cos α
- Tan (2π – α ) = -tan α
- Cot (2π – α ) = – cot α

4 . Complementary Angles (= sum is π/2)

- Sin ( π/2 – α ) = cos α
- Cos (π/2 – α ) = sin α
- Tan (π/2 – α ) = cot α
- Cot (π/2 – α ) = tan α

## Trigonometry Angles Table

Here there are some special angles provided with the trigonometric numbers. To simplify the way of calculation of the trigonometric numbers at various angles, reference angles are used which are derived from the primary trigonometric functions. We can derive values in degrees like 0^{0}, 30^{0}, 45^{0}, 60^{0}, 90^{0}, 180^{0 }, 270^{0} and 360^{0}. The trigonometric table of all angles are given below, which defines all the values of trigonometric ratios.

## Worked-out Problems:

### Question:

If sin 3A = cos (A-26^{0}), where 3A is an acute angle, find the value of A.

### Solution:

Given that, sin 3A = cos (A-26^{0}) ….(1)

Since, sin 3A = cos (90^{0} – 3A), we can write (1) as

cos(90^{0}– 3A)= cos (A- 26^{0})

Since , 90^{0}-3A = A – 26^{0}

Therefore,

90^{0} + 26^{0} = 3A + A

4 A = 116^{0}

A = 116^{0} / 4 = 29^{0}

Therefore the value of A is 29^{0}.

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