The three most familiar trigonometric ratios in the trigonometric functions 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°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. Here, let us take the discussion for the value for cos 90 degrees which is equal to zero and how the values are derived using the quadrants of a unit circle.
Value of Cos 90° = 0
Cos 90 degrees
To define the cosine function of an acute angle, consider a right-angled 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 θ = Adjacent Side / Hypotenuse Side
Derivation to Find Cos 90 Degrees Value Using Unit Circle
Let us consider a unit circle with the center at the origin of the coordinate axes say ‘x’ and ‘y’ 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 right-angled triangle OMP
By using the Pythagorean theorem, we get
OM2+ MP2= OP2 (or) a2+ b2= 1
Thus, every point on the unit circle is defined as
a2+ b2 = 1 (or) cos2 x + sin2 x = 1
Note that the one complete revolution subtends an angle of 2π radian at the center of the circle, and from the unit circle it is defined as follows:
∠AOC = π and
Since all angles of a triangle are the integral multiples of π/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), (–1, 0) and (0, –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
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 2π. 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
c2 = a2 + b2– 2ab cos(C)
Also, it is easy to remember the special values like 0°, 30°, 45°, 60°, and 90° since all the values are present in the first quadrant. All the sine and cosine functions in the first quadrant take the form √(n/2) or √(n/4). Once we find the values of sine functions it is easy to find the cosine functions.
Sin 0° =√(0/4)
Sin 30° = √(1/4)
Sin 45° = √(2/4)
Sin 60° = √(3/4)
Sin 90° = √(4/4)
Now Simplify all the sine values obtained and put in the tabular form:
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 0° = Sin 90°
Cos 30° = Sin 60°
Cos 45° = sin 45°
Cos 60° = sin 30°
Cos 90° = sin 0°
So the value of cos 90 degrees is equal to 0 since cos 90° = sin 0°
In a similar way, we can find the values of other degrees of trigonometric functions depends on the quadrant value.
Find the value of cos 135°
Cos 135°= cos(90°+145°)
Now, take the values a = 90° and b = 45°
By using the formula, Cos(a+b) = cos a cos b – sin a sin b
So, it becomes Cos 135° = cos 90° cos 45° +sin 90° sin 45°
cos 135°=0\1/√2 – 1\1/√2
Get more information on cos 90 degrees and other trigonometric functions, visit BYJU’S and also watch the interactive videos to clarify the doubts.