What Is the Value of Cos 90 Degrees in Trigonometry

Trigonometry is one of the most important mathematical tools used for day-to-day measurements. Trigonometry is used to measure the angles and sides of a triangle which are not known. However this holds only for triangles  which are right angled. It uses ratios and reciprocals to find the measurements of unknown values. The three main components of trigonometry are Sin, cosine and tangent, based on which all calculations and measurements are done. Cosine or cos is one of those important trigonometric ratios which is actually the ratio of the lengths of adjacent sides and hypotenuses. To know about these important trigonometric functions we start here with the value of Cos 90 degree, the laws relevant to it, the reciprocals and also the derivations relevant to it with the help of unit circle. The value of cos 90 degrees has been derived to be 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:

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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.

\[Cos Ө = \frac{\text{Adjacent Side}}{\text{Hypotenuse side}} \]

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,

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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

Cos2  x + 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:

\[\angle AOB = \frac{\pi}{2}\]

\[\angle AOC = \pi\] and  \[\angle AOD = 3\frac{\pi}{2}\]

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

\[c^{2} = a^{2} + b^{2} - 2ab Cos (c) \]

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  \[\sqrt{\frac{n}{2}}\] or \[\sqrt{\frac{n}{4}}\].

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

Sin 0°   = \[\sqrt{\frac{0}{4}}\]. 

Sin 30° = \[\sqrt{\frac{1}{4}}\].

Sin 45° = \[\sqrt{\frac{2}{4}}\].

Sin 60° = \[\sqrt{\frac{3}{4}}\].

Sin 90° = \[\sqrt{\frac{4}{4}}\]. 

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

Cos 0° =Sin 90° =1  

Cos 30° = Sin 60°  =   \[\sqrt{\frac{3}{4}}\] =  \[\frac{\sqrt{3}}{2}\]

Cos 45° = Sin 45 ° =  \[\sqrt{\frac{2}{4}}\] = \[\frac{1}{\sqrt{2}}\]

Cos 60° = Sin 30°  = \[\sqrt{\frac{1}{4}}\] = \[\frac{1}{2}\]

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. \[ 4 (Sin^{2}30° + Cos^{2}60°) – 3(Cos^{2}45° - Sin^{2} 90°)\]

Solution- Value of Sin 30° =\[\frac{1}{2}\]

Value of Cos 60° = \[\frac{1}{2}\]

Value of Cos 45° = \[\frac{1}{\sqrt{2}}\]

Value of Sin 90° = 1

= \[4 ( (\frac{1}{2})^{2} + (\frac{1}{2})^{2} ) – 3 ((\frac{1}{\sqrt{2}})^{2} -1^{2}) \]

=\[4 (\frac{1}{4} + \frac{1}{4}) - 3(\frac{1}{2} -1) \]

=\[ (4 \times \frac{2}{4}) - (3 \times \frac{-1}{2}) \]

= \[ (4 \times \frac{1}{2}) - ( \frac{-3}{2}) \]

=\[ (\frac{4}{2}) + ( \frac{3}{2}) \]

=\[ \frac{4+3}{2} \]

=\[ \frac{7}{2} \]

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 Ө

= Sin Ө = (\[\sqrt{2}\] -1) Cos Ө

(Multiply with both side ‘\[\sqrt{2}\] + 1 ‘)

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

= \[\sqrt{2}\] Sin Ө + Sin Ө = (\[\sqrt{2}^{2}\] - 1) Cos Ө = (2 - 1) Cos 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