Cosec 240

Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a correct-angled triangle. Trigonometric functions, also known every bit goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The vi primary trigonometric functions are sine, cosine, tangent, cotangent, secant, or cosecant.

Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.

As given in the above figure in a right-angled triangle:

• Hypotenuse:
  The side opposite to the right angle is the hypotenuse, It is the longest side in a correct-angled triangle and opposite to the 90° bending.
• Base of operations:
  The side on which angle C lies is known every bit the base.
• Perpendicular:
  It is the side contrary to angle C in consideration.

Trigonometric Functions

Trigonometry has 6 basic trigonometric functions, they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions. The six trigonometric functions are as follows:

• sine:
  It is defined every bit the ratio of perpendicular and hypotenuse and Information technology is represented as sin θ
• cosine:
  It is defined as the ratio of base and hypotenuse and it is represented as cos θ
• tangent:
  Information technology is divers equally the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base and is represented as tan θ
• cosecant:
  It is the reciprocal of sin θ and is represented as cosec θ.
• secant:
  It is the reciprocal of cos θ and is represented equally sec θ.
• cotangent:
  It is the reciprocal of tan θ and is represented as cot θ.

According to above given paradigm, Trigonometric Ratios are

• Sin θ
  = Perpendicular / Hypotenuse = AB/AC
• Cosine θ
  = Base / Hypotenuse = BC / AC
• Tangent θ
  = Perpendicular / Base of operations = AB / BC
• Cosecant θ
  = Hypotenuse / Perpendicular = Air conditioning/AB
• Secant θ
  = Hypotenuse / Base = AC/BC
• Cotangent θ
  = Base / Perpendicular = BC/AB

Reciprocal Identities

Sin θ = 1/ Cosec θ                    Or        Cosec θ = 1/ Sin θ

Cos θ = 1/ Sec θ                       Or         Sec θ = 1 / Cos θ

Cot θ = ane / Tan θ                     Or         Tan θ = 1 / Cot θ

Cot θ = Cos θ / Sin θ               Or          Tan θ = Sin θ / Cos θ

Tan θ.Cot θ = 1

Values of Trigonometric Ratios

	0°	thirty°	45°	60°	90°
Sin  θ		1/two	1/√two	√3/2	1
Cos  θ	1	√3/2	1/√2	ane/2
Tan  θ		1√3	1	√3	Not Defined
Cosec θ	Non Defined	2	√2	2/√iii	1
Sec  θ	1	2/√3	√2	2	Not Defined
Cot  θ	Non Defined	√3	1	ane/√iii

Trigonometric Identities of Complementary and Supplementary Angles

• Complementary Angles:
  Pair of angles whose sum is equal to 90°
• Supplementary Angles:
  Pair of angles whose sum is equal to 180°

Identities of Complementary angles are

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

Identities of supplementary angles

sin (180° – θ) = sin θ

cos (180° – θ) = – cos θ

tan (180° – θ) = – tan θ

cot  (180° – θ) = – cot θ

sec (180° – θ) = – sec θ

cosec (180° – θ) = – cosec θ

Quadrants of trigonometry

Notice the value of sin 120° cosec 240° tan 210°

Solution:

Here we accept sin 120° cosec 240° tan 210°

We can write as Sin (ninety+30) cosec (180+60) tan (180+30)  { equally per the quadrants and trigonometric values }

=  Sin 30° × Cosec sixty° × ( -Tan 30°)

=  1/2 × 2/√iii × -1√iii

= – i/3

The value of sin 120° cosec 240° tan 210° is – 1/3

Like Questions

Question 1: What is the exact value of sin 270?

Solution:

Here sin is positive only in 1st and 2nd Quadrant.

270° does non lies in 1st and 2d Quadrant.

Therefore     sin (360° – θ) = – sin θ

                          sin (270°) = sin (360° – 90°)

                          sin (270°) = – sin (90°)

                          sin (270°) = – 1

So the exact value of sin 270 is -i

Question 2: Evaluate (Sin 30° – Sin 90° + 2 Cos 0°) / Tan 30° Tan threescore°?

Solution:

Here we have (Sin 45° – Sin ninety° + 2 Cos 0°) /  Tan 45° Tan 60°

Every bit per the trigonometric values

(Sin 45° – Sin xc° +2 Cos 0°) /  Tan 45° Tan 60°

= (1/√2 – 1 + 2 × 1) /  i × √3

= (1/√2 – 1 + 2) / √3

= (1/√ii + 1) / √three

= (1+√2 / √2) / √iii