The most common trigonometric ratios that help in trigonometric functions are cosine, sine and tangent function. This is generally defined as angles which are less than a right angle. Even in the trigonometric functions, the ratio of two sides of a right-angled triangle which has an angle with values found in multiple line segments. It is linked to the length of line segments surrounding a unit circle.
Ideally, 0°, 45°, 30°, 60°, 180°, 90°, 270° and 360° are a form of representation of degree. Students in this segment can learn the value of cos 90 degrees whose value is often equal to zero. It is similar to the way the values are derived using a unit circle’s quadrants.
It is represented as the value of Cos 90° = 0.
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.
The hypotenuse side can be defined as the opposite side of a right angle. This side is usually the longest in a right triangle
The opposite to the angle of interest is also the opposite side in a triangle.
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.
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’.
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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 is given as (0, 1), (1, 0), (–1, 0) and (0, –1). This is also known as quadrant angles and coordinates, which makes the value of cos 90 degrees to be zero.
It can be 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 be 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).
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1. What is the Answer of the Equation 6(Sin²30⁰ + cos²60⁰) - 4(cos² 45⁰ - Sin²90⁰)?
Ans. Here the value of sin 30⁰ is 1/2 while the value of cos 60⁰ is ½
Moreover, the value of cos 45⁰ is (1/ √2)
Value of sin 90⁰ is 1 in the equation
The solution will be 5 [(½)²+(½)² ]- 4[(1/ √2)²-1²]
We take 6[1/4+1/4 ]- 4[1/2-1]
This gives us 3 x2/4-4(1/2-1)
Now taking 3x1/2-4/2-1
The value will be 1.5-4(-1/2)
This gives us 1.5-4/2
The final answer will -1/2
2. What Does Inverse Cosine Function Mean?
Ans. Inverse Cosine is a form of Trigonometric functions. These functions have a contrary part to it, irrespective of cosecant, secant, sine, tangent, cosine, and cotangent. These functions can be used to solve multiple mathematical equations apart from trigonometric formulas.
Inverse functions are also called as Arc Functions as they provide an arc’s length belonging to a trigonometric function’s value. Ideally, they are written as arccos(x), arcsin(x) and arctan(x). Cosine function in a right-angled triangle is the ratio of the base length of the adjacent side to that of the longest side of a triangle. Here an Inverse Cosine functions as the inverse as it is used to attain a right-angled triangle’s value of the angle.
3. What are the Properties of Cosine in Terms of Quadrants?
Ans. The first quadrant is said to lie between 0⁰ and 90⁰ angles. Angles between 180⁰ and 270⁰ are in a third quadrant while a second quadrant falls between 90⁰ and 180⁰. The fourth quadrant lies between 270⁰ and 360⁰.
Cos remains positive in a fourth quadrant. It is important to note that all are positive in the first one. In the second Sin and third quadrant Tan is also positive. An example will be the distance from a point to the origin remains positive, but there X and Y coordinates can be either negative or positive.
Thus, in a first quadrant, all coordinates are positive, while only sine and cosecant are positive in a second quadrant. Tangent and cotangent remain only positive in a third quadrant while cosine and secant in a fourth one.