 # Cos 90 Value

## Value of Cos 90 Degree

Sine, Cosine, and Tangent are the three primary trigonometric functions through which trigonometry formulas and trigonometry ratios are defined. In a right-angled triangle, the cosine function of an acute angle is stated as the ratio of the length of the adjacent side of a right-angle triangle and hypotenuse. Here in this article, we will discuss the value of Cos 90 degree, the law of cosine, inverse cosines, and how to derive cos 90 value using unit circle.

### What is the Value of Cos 90?

The value of Cos 90 degree is 0

Cos 90 = 0

### Cos 90 Value

To define the cosine function of an acute angle, we will consider the right angle triangle with an angle of interest and the sides of a triangle. The three sides of a triangle are defined as:

The opposite side of a right-angle triangle is the side opposite to the angle of interest.

The longest side of a right-angle triangle i.e. hypotenuse is the opposite side of a right angle in the triangle.

The remaining side of a triangle i.e. an adjacent side forms between both the angle of interest and the right angle.

The cosine function of an angle is defined as the ratio of the length of the adjacent side of a right-angle triangle and hypotenuse.

Cos Ө = Adjacent Side / Hypotenuse side

### Derivation of Cos 90 Degrees Using Unit Circle

Now, we will calculate the value of cos 90° using the unit circle with the radius 1 unit and the center of the circle placed at the origin of the coordinate axis ‘x’ and ‘y’. Let us take the point P (a,b) anywhere in the circle that forms an angle AOP= x radian. It implies that the length of the arc AP is equivalent to x. With this, we will define the following value,

Cos x= a and Sin x =b.

With the unit circle, we will now consider the right angle triangle OMP.

Through Pythagorean Theorem, we get

OM2 + MP2 = OP2

or

a2 + b2 = 1

Thus, each point on the unit circle is defined as

A2 +B2 = 1

or

Cosx + Sin2 x = 1

Note- One complete revolution subtends an angle of 2 π radian at the center of the circle, and from the unit circle. It is defined as:

∠AOB = π/2

∠AOC = π and

∠ AOD =3 π/2

All the angles of a triangle are the integral multiples of π/2 and it is usually known as quadrant angle. The coordinates of point A, B, C and D are stated as (1, 0), (0, 1), (-1,0) and (0.-1) respectively. We will get the Cos 90 value through the quadrant angle.

Therefore, the value of Cos 90° is 0

Cos 90 = 0

It can be seen that the value of the sine and cosine function does not change if the x and y values are the integral multiples of π/2. If we will consider one complete revolution from the point P, it will come back again to the same point. For triangle ABC, with sides a, b, and c opposite to the ∠A, ∠B, and ∠C respectively, the cosine law will be defined.

With ∠C, the law of sine is stated as

C2 =+a2 + b² - 2ab Cos (c)

With this, it will be easy to remember 0°,30°, 45° 60° and,90° as all these values are present in the first quadrant. Each sine and cosine function in the first quadrant takes the form $\sqrt{(n/2)}$ or $\sqrt{(n/4)}$.

We can easily find the value of the cosine function if we know the values of the sine function.

Sin 0°   =$\sqrt{(0/4)}$

Sin 30° =$\sqrt{(1/4)}$

Sin 45° =$\sqrt{(2/4)}$

Sin 60° =$\sqrt{(3/4)}$

Sin 90° =$\sqrt{(4/4)}$

Now through the sine value, we can find the cosine value easily because

Cos 0° =Sin 90° =1

Cos 30° = Sin 60°  = $\sqrt{(3/2)}$

Cos 45° = Sin 45 ° =$\sqrt{(1/2)}$

Cos 60° = Sin 30°  = 1/2

Cos 90° = Sin 0° = 1

So accordingly, the value of Cos 90 degree or Cos 90 =0

Similarly, values of other degrees of trigonometry functions can be found out.

### Solved Example

1. Evaluate the following

a.  4 (Sin2 30° + Cos260°) – 2(Cos2 45° - Sin2 90°)

Solution- Value of Sin 30° =1/2

Value of Cos 60° = ½

Value of Cos 45° = (1/$\sqrt{2}$)

Value of Sin 90° = 1

= 4[(½)2 + (½)2] – 3 [(1/$\sqrt{2}$)2 -12]

=4[¼ + ¼] -3[½ -1]

= 2 x 2/4 – 3(1/2- 1]

= 2x ½ -3/2 -1

=1-3(-1/2)

=1-3/2

=-½

b. If Cos Ө or Sin Ө =$\sqrt{2}$Cos Ө, show that Cos Ө –Sin Ө =$\sqrt{2}$Sin Ө.

Solution: Cos Ө or Sin Ө =$\sqrt{2}$Cos Ө

= Sin Ө =$\sqrt{2}$Cos Ө – Cos

= Sin Ө = ($\sqrt{2}$ -1) Cos Ө

= ($\sqrt{2}$ + 1) Sin Ө = ($\sqrt{2}$ + 1) ($\sqrt{2}$ - 1) Cos Ө

= $\sqrt{2}$ Sin Ө + Sin Ө = Cos Ө

= $\sqrt{2}$ Sin Ө = Cos Ө –Sin Ө

Hence, Proved

### Fun Facts

In trigonometry, the law of cosines is also known as cosine formula, cosine rule or al-Kashi’s theorem.

The position of  the ship can be determined through trigonometry and Marine chronometer.

Hipparchus, who compiled the first trigonometry table, is also known as “Father of Trigonometry”.

### Quiz Time

1. For any acute angle, cosine would be equal to

a. –Cos (180°- Ө)

b. Cos (180° - Ө)

c. –Cos (180° + Ө)

d. Cos (180° +Ө)

2.  The Cosine Rule is also known as

a. Sine triangle

b. Cosine Triangle

c. Cosine Area

d. Cosine Formula

3. Trigonometry is based on

a. Squares

b. Rectangle

c. Triangle

d. Octagons

Explain Inverse Cosine Function

The Inverse cosine function is used to calculate the unknown value of an angle if the adjacent side or base and hypotenuse values are given in the question. The inverse Cosine is also known as Arcos and is denoted as Cos-1.

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For a right triangle with adjacent side √3, Hypotenuse 2 cm and Perpendicular= 1 cm

Value of Cos A = Adjacent Side/ Hypotenuse

Accordingly,

Cos (a) = √3 /2

Now angle a will be calculated using Cos-1 or inverse cosine function.

∠a = Cos-1 (√3/2)

∠a = π/6 = 30°.

Explain Cosine Properties in Terms of its Quadrants?

It is essential to know that cosine value changes with respect to the quadrants. The Cosine values which lie in the first and fourth quadrants have positive values and the values which lie in the second and third quadrants have negative values. For example, Cos 120°,150°, and 180° have positive values as they lie in the second quadrant while Cos 0° and 30° have negative values as they lie in the first quadrant.