# Trigonometry Angles

Trigonometry angles are the angles given by the ratios of the trigonometric functions. Trigonometry deals with the study of the relationship between angles and the sides of a triangle. The angle value ranges from 0-360 degrees. The important angles in trigonometry are 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. And the important six trigonometric ratios or functions are sine, cosine, tangent, cosecant, secant and cotangent. Before discussing the trig angles, let us have a look at the definition of angle, and its related terminologies. Trigonometry is an important topic for Class 10, 11 and 12. Hence, it is required for students to learn the concept in a detailed manner to excel in exam. Learn all the related trigonometric topics here with us at BYJU’S.

## Trigonometry Angles Table (0° to 360°)

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°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. The trigonometric table of all angles is given below, which defines all the values of trigonometric ratios.

 Angle (in Degrees) 0° 30° 45° 60° 90° 180° 270° 360° Angle (in Radians) 0 π/6 π/3 π/4 π/2 π 3π/2 2π Sin 0 1/2 1/√2 √3/2 1 0 -1 0 Cos 1 √3/2 1/√2 1/2 0 -1 0 1 Tan 0 1/√3 1 √3 ∞ 0 ∞ 1 Cot ∞ √3 1 1/√3 0 ∞ 1 ∞ Sec 1 2/√3 √2 2 ∞ -1 ∞ 1 Cosec ∞ 2 √2 2/√3 1 ∞ -1 ∞

### Special Angles

The special angles used in trigonometry are 0°, 30°, 45°, 60° and 90°. These are the common angles which are used while performing computations of trigonometric problems. Hence, it is suggested for students to memorise the values of trigonometric ratios (sine, cosine and tangent) for these angles, to do quick calculations.

### Positive and Negative Angles

The angles could be positive or negative in nature. If the angle is formed in a counterclockwise direction from the start point in an x-y plane, then it is positive and if the angle is formed in a clockwise direction from the start point then it is a negative angle.

### Angles More Than 360°

Once a complete cycle is done in a an x-y plane, that means if we start the cycle from 0° and ends at 360°, a full circle is made. One full cylce also resembles a unit circle. After that, if we still keep going then the angles are greater than 360°. Now beyond 360° at each quadrant, you will get the angles, such as, 450°, 540°, 630°, 720°, and so on. At 720° two cycles are completed and with respect to radians it is measured as 4π. Similarly, the value of radians at each cycle is increased as n x 2π. So we can show this in a table as:

 Number of Cycle Angle (n x 360°) Radians (n x 2π) 1 360° 2π 2 720° 4π 3 1080° 6π 4 1440° 8π

## 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 :

### 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

### 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

### 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

### 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

## Angle Definition

The two rays that have the same beginning point that forms the figure called 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 the 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 900. 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 the radius. Because the length of the full circle is 2πr. Alternatively, we can say that the circle contains 2π radians. The subdivision of radians is also written in decimal forms.

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

Because 2 π = 360°, following conversion formulas has to be applied.

• r rad → [ 360.r/2π
• g°→ [ 2π .g/360] rad

## Trigonometric Angles Questions

Question 1: Find the value of sin 60 – cos 30.

Solution: The value of sin 60 = √3/2

Value of cos 30 = √3/2

Hence, sin 60 – cos 30 = √3/2 – √3/2 = 0

Question 2: Evaluate the value of 3sin 30 + tan 45

Solution: Value of sin 30 = 1/2

Value of tan 45 = 1

By putting the values we get;

=3(1/2)+1

=3/2+1

=5/2

Question 3: If sin 3A = cos (A-26°), where 3A is an acute angle, find the value of A.

Solution:

Given that, sin 3A = cos (A-26°) ….(1)

Since, sin 3A = cos (90° – 3A), we can write (1) as

cos(90°- 3A)= cos (A- 26°)

Since , 90°-3A = A – 26°

Therefore,

90° + 26° = 3A + A

4 A = 116°

A = 116° / 4 = 29°

Therefore the value of A is 29°.

Question 4: If α = 60° and β = 30°, prove that sin (α – β) = sin α cos β – cos α sin β.

Solution: L.H.S. = sin (α – β)

= sin (60° – 30°)

= sin 30°

= ½

R.H.S. = sin α cos β – cos α sin β

= sin 60° cos 30° – cos 60° sin 30°

= 3√2×3√2−12×12

= ¾ – ¼

= 2/4

= ½

Hence, L.H.S. = R.H.S. (Proved)

Question 5: If α = 30°, verify that 3 sin α – 4 sin3 α = sin 3α.

Solution: L.H.S = 3 sin α – 4 sin3 α

= 3 sin 30° – 4. sin3 30°

= 3 ∙ (1/2) – 4 ∙ (1/2)3

= 3/2 – 4 ∙ 1/8 3/2 – ½

= 1

R.H.S. = sin 3α

= sin 3 ∙ 30°

= sin 90°

= 1

Hence, L.H.S. = R.H.S. (Proved)

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