The trigonometric functions are also known as an angle function that involves the study of triangles where it relates the angles of a triangle and the dimension of the triangles. The study of trigonometric functions plays an important role in our career and it is applied in various fields like engineering and architecture fields and also in the study of periodic phenomena like sound and light waves, and temperature variations. The three most familiar trigonometric ratios are sine function, cosine function, and tangent function. It is generally defined for the angles less than a right angle, and the trigonometric functions are stated as the ratio of two sides of a right triangle containing the angle in which the values can be found in the length of various line segments around a unit circle. Usually, the degrees are represented in the form of 0^{0}, 30^{0}, 45^{0}, 60^{0}, 90^{0}, 180^{0}, 270^{0} and 360^{0}. Here, let us take the discussion for the value for cos 90 degrees and how the values are derived using the quadrants of a unit circle.
Cos 90 degrees
To define the cosine function of an acute angle, consider a rightangled triangle provided with the angle of interest and the sides of a triangle. The three sides of the triangle are defined as follows:
 The opposite side is a side which is opposite to the angle of interest.
 The hypotenuse side is the opposite side of the right angle and it should be the longest side of a right triangle
 The adjacent side is the remaining side of a triangle where it forms a side of both the angle of interest and the right angle
The cosine function of an angle is defined as a ratio of the length of the adjacent side to the length of the hypotenuse side and the formula is given by
Cos Î¸ = Adjacent Side / Hypotenuse Side
Derivation to Find Cos 90 Degree Value Using Unit Circle
Let us consider a unit circle with the center at the origin of the coordinate axes say â€˜xâ€™ and â€˜yâ€™ axis. Let P (a, b) be any point on the circle that forms an angle AOP = x radian. This means that the length of the arc AP equals to x. From this, we define the value that cos x = a and sin x = b.
By using the unit circle, consider a rightangled triangle OMP
By using the Pythagorean theorem, we get
OM^{2}+ MP^{2}= OP^{2} (or) a^{2}+ b^{2}= 1
Thus, every point on the unit circle is defined as
a^{2}+ b^{2} = 1 (or) cos^{2} x + sin^{2} x = 1
Note that the one complete revolution subtends an angle of 2Ï€ radian at the center of the circle, and from the unit circle it is defined as follows:
âˆ AOB=Ï€/2,
âˆ AOC = Ï€ and
âˆ AOD =3Ï€/2.
Since all angles of a triangle are the integral multiples of Ï€/2 and it is commonly known as quadrantal angles and the coordinates of the points A, B, C and D are given as (1, 0), (0, 1), (â€“1, 0) and (0, â€“1) respectively. We can get the cos 90 degrees value using the quadrantal angle. Therefore, the value of cos 90 degrees is
Cos 90^{0} = 0
It is observed that the values of sin and cos functions do not change if the values of x and y are the integral multiples of 2Ï€. When we consider the one complete revolution from the point p, it again comes back to the same point. For a triangle, ABC having the sides a, b, and c opposite the angles A, B, and C respectively, the cosine law is defined.
For an angle C, the law of cosine is stated as
c^{2 }= a^{2 }+ b^{2}– 2ab cos(C)
Also, it is easy to remember the special values like 0Â°, 30Â°, 45Â°, 60Â°, and 90Â° since all the values are present in the first quadrant. All the sine and cosine functions in the first quadrant take the form \(\frac{\sqrt{n}}{2}\) or Â \(\sqrt{\frac{n}{4}}\) . Once we find the values of sine functions it is easy to find the cosine functions.
Sin 0Â° = \(\sqrt{\frac{0}{4}}\)
Sin 30Â° = \(\sqrt{\frac{1}{4}}\)
Sin 45Â° = \(\sqrt{\frac{2}{4}}\)
Sin 60Â° = \(\sqrt{\frac{3}{4}}\)
Sin 90Â° = \(\sqrt{\frac{4}{4}}\)
Now Simplify all the sine values obtained and put in the tabular form:
0Â° 
30Â°  45Â°  60Â° 
90Â° 

Sin 
0 
1/2  \(\frac{1}{\sqrt{2}}\)  \(\frac{\sqrt{3}}{2}\) 
1 
From the values of sine, we can easily find the cosine function values. Now, to find the cos values, fill the opposite order the sine function values. It means that
Cos 0Â° = Sin 90Â°
Cos 30Â° = Sin 60Â°
Cos 45Â° = sin 45Â°
Cos 60Â° = sin 30Â°
Cos 90Â° = sin 0Â°
So the value of cos 90 degrees is equal to 0 since cos 90Â° = sin 0Â°
0Â° 
30Â°  45Â°  60Â° 
90Â° 

Sin 
0 
1/2  \(\frac{1}{\sqrt{2}}\)  \(\frac{\sqrt{3}}{2}\) 
1 
Cos 
1 
\(\frac{\sqrt{3}}{2}\)  \(\frac{1}{\sqrt{2}}\)  1/2 
0 
In the similar way, we can find the values of other degrees of trigonometric functions depends on the quadrant value.
Sample Example
Question:
Find the value of cos 135^{0}
Solution:
Cos 135^{0}= cos(90^{0}+145^{0})
Now, take the values a = 90^{0} and b = 45^{0}
By using the formula, Cos(a+b) = cos a cos b – sin a sin b
So, it becomes Cos 135^{0} = cos 90^{0} cos 45^{0} +sin 90^{0} sin 45^{0}
\(\cos 135^{\circ}=0\frac{1}{\sqrt{2}}1\frac{1}{\sqrt{2}}\) \(\cos 135^{\circ}=\frac{1}{\sqrt{2}}\)Get more information on cos 90 degrees and other trigonometric functions, visit BYJUâ€™S and also watch the interactive videos to clarify the doubts.