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# Cos 90 Value in Maths

Sine, tangent and cosine are the primary trigonometric functions in mathematics. Through these three, all the trigonometric formulas and identities have been defined.

Cosine is the ratio of the length of the adjacent side of the right-angled triangle, i.e. the base, to the hypotenuse.

The value of cos 90 is 0. We can see from the graph, the value of cos 90 reaches 0 on the x-axis.

We know, cos = base/hypotenuse

From the right-angled triangle XYZ, taking angle c is the reference,

Cos c = YZ/XZ

Now, taking angle a as the reference,

Cos a = XY/XZ

## Derivation of cos 90 using the unit circle method

Consider a unit circle with the centre as the origin. The radius of the circle is 0.5 units and is located on the x and y axes.

The coordinates of OA are (1,0), OB (0,1), OC (-1,0) and OD (0,-1)

Take a triangle inside the circle as shown in the figure. OP is the hypotenuse of that triangle (radius of the circle) with a unit length.

The length of OM is a, and PM is b.

From the Pythagoras theorem, in the right-angled triangle OPM,

OM² + PM² = OP²

This implies, a²+ b² = 1

Thus, every point on the unit circle will be denoted as;

a²+ b² = 1

cos² x + sin² x = 1

We know that one complete revolution subtends an angle of 2π radian. From the unit circle it is written as-

∠AOB=π/2,

∠AOC = π

∠AOD =3π/2

We know that all the angles of a triangle are integral multiples of π/2. Therefore, from the quadrant angle, we get Cos 90° = 0.

## List of trigonometric ratios

 Angles (In Degrees) 0 30 45 60 90 180 270 360 Angles (In Radians) 0 π/6 π/4 π/3 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 Not Defined 0 Not Defined 0 cot Not Defined √3 1 1/√3 0 Not Defined 0 Not Defined cosec Not Defined 2 √2 2/√3 1 Not Defined −1 Not Defined sec 1 2/√3 √2 2 Not Defined −1 Not Defined 1

## Example

Find the value of cos 135°

## Solution

Cos 135° can be written as cos (90°+45°)

From the formula, Cos (a+b) = cos a cos b – sin a sin b

Cos 135° = cos 90° cos 45° – sin 90° sin 45°

cos 135°=0 x 1/√2 – 1 x 1/√2

cos 135°=-1/√2

## Do you know?

1. Hipparchus is known as the father of trigonometry.
2. With the law of cosines, one can find the values of a non-right-angled triangle. The Pythagorean Theorem is a special case of the law of cosines.
3. Cos values have a significant role in radio waves, musical tones, electrical currents and tides. They are very important in the life of an architect, surveyor, astronaut and crime scene investigator.
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