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How to Find Cos 90 Degrees Value Using Unit Circle?
The cosine of 90 degrees is 1. To find the value of cosine 90 degrees on a unit circle. One way is to use the Pythagorean theorem. The length of the hypotenuse of a right triangle is the square root of the sum of the squares of the other two sides. In this case, the other two sides would be the length of the triangle’s base and its height. So, the length of the hypotenuse would be the square root of the sum of the squares of 9 and 10, which is 11. Therefore, cos90 degrees equals 11/12. Another way to evaluate cos90 degrees is to use the inverse cosine function. The inverse cosine function is written as cos-1. So, the inverse cosine of 11/12 would be cos-1(11/12), which is about 0.766.
The Way to Evaluate Cos 90 Degrees
To define the cos 90-degree function of an acute angle, students can take a right-angled triangle which has the sides of a triangle and an angle of interest. The sides of the triangle can be defined by the following method.
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The hypotenuse side can be defined as the opposite side of a right angle. This side is usually the longest in a right triangle
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The opposite to the angle of interest is also the opposite side in a triangle.
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The adjoining side of a triangle is its remaining side of a triangle which forms a side of the right angle and the angle of interest.
The formula Cos θ = Adjacent Side divided by Hypotenuse Side gives the hypotenuse side of a cosine function. Usually, the ratio of the length of the adjoining side to the size is the cosine function of an angle.
How to Find Cos 90 Degrees Value Using Unit Circle?
To understand the concept better let’s take an example where a unit circle at the centre origin of the coordinate axes x and y. Here AOP forms an x radian which is linked to a point p (a and b). This makes AP equal to x which is a length of the arc. By solving this example, we can find the value of sin x to be ‘b’ and cos x as ‘a’.
Now take a right-angle triangle OMP employing a unit in a circle. Here if we take Pythagorean theorem, it gives a2+ b2= 1
We can also take OM2+ MP2= OP2
By using this formula, we find all point on a unit circle to be a2+ b2 = 1 (or) cos2 x + sin2 x = 1. It is important to know that angle of 2π of cos 90 radians forms one revolution. This complete cycle also subtends at the centre of a circle.
We can define as ∠AOC = π, ∠AOD =3π/2 and ∠AOB=π/2
Finally, we can get the cos 90 degrees in radians value using the quadrant angle. This is possible as all angles in a triangle are integral multiples of π/2. Coordinates B, A, C and D here given as (0, 1), (1, 0), (–1, 0) and (0, –1). This also known as quadrant angles and coordinates, which makes the value of cos 90 degrees to zero.
It can seen that values of cos and sin functions don’t affect the values of x and y, which is 2π’s integral multiple. From point p, the complete change or revolution gets back to the same point. The sides a, b and c of a triangle ABC is also opposite of A, B and C as per cosine law. While for C angle, this becomes c2 = a2 + b2– 2ab cos(C).
To find the values of sine functions by calculating the cosine of a 90-degree triangle, students must remember values like 0°, 45°, 30°, 60°, and 90°. These values are present in the first quadrant where sine and cosine functions form √(n/4) or √(n/2).
The following can used by students that is Sin 45° = √(2/4), Sin 60° = √(3/4), Sin 0° =√(0/4), Sin 90° = √(4/4) and Sin 30° = √(1/4).
