Cosine Rule
What is the Meaning of Cos?
Cos is a primary function related to the sums of Trigonometry and calculus. It is simply the cosine of a specific angle of a triangle. The cos of different angles has their specific values which play a vital role in the case of solving trigonometric problems. In calculus, it relates to the Sin function and together they make formulas and identities. There are many concepts of trigonometry (conceptual geometry) and you need to learn them thoroughly to get good marks in the examination.
Cosine: Definition
As per basic trigonometry is concerned, it deals with various operations with the right-angled triangle. Different denotations are made according to the relations of each line segment of the triangle and its relation with the other. These line segments have specific names such as the Opposite (Height of the triangle), Adjacent (base of the triangle), and Hypotenuse (the line joining the tip of the base to height).
The relations of the line segments of the triangle are expressed with three specific functions known as Sine, Cosine, and Tangent. In the case of the sums, these terms are abbreviated as sin, cos, and tan.
Here, you will get to know about Cosine. If you consider a right-angled triangle, the angle made by the base and hypotenuse will be considered as θ. In the case of the Cosine function, the angle between the adjective and hypotenuse is only taken into account.
So, cos θ = Base / hypotenuse
The Cos Formula
The details of the Cos formula is mentioned below:
(Image will be uploaded soon)
In reference to the above picture you can say that in the Triangle ABC, Cos Alpha can be written as:
Cos α = The dividend of the side AC and AB that is AB/AC
So, Cos α = b/h (Which matches the concept totally)
Tables and Identities of Cosine
As a student learning trigonometry, you must keep the tables related to the Cosine function in mind.
Generally, the values of different angles are given in the table. For the basic level trigonometry, you will only need to know the values till 90 degrees starting from 0 degrees. You can refer to the trigonometry study materials from Vedantu to know all the values. Learning these values will help you to understand what is the meaning of cos.
If you take mathematics in higher studies, you might have to learn the values of Cosine angles up to 360 degrees.
Apart from the values of different angles, you need to learn the different identities of Cosines. As the identities of Cosine can help you to solve problems easily. As Cosine is a primary function it is mainly related to Sine (sin). All the identities of Cosine are actually formulas of Cosine that can help you to jump steps accordingly. All you need to do is memorize the identities thoroughly.
Law of Cosine
The Law of Cosine is the better way to know about Cosine. The relationship between the sides of the right-angled triangle is mathematically displayed here the law is given below:
c2 = a2 + b2 + 2ab cos α
In the above formula, variables a, b, and c denote the three sides of the triangle. To know more about the derivation of the Cosine formula you need to visit the Vedantu site.
Why is the Concept of Cosine so Vital?
The concept of Cosine is vital because it is one of the primary trigonometric functions that you need to know. Without knowing all about Sine, Cosine, and Tangent, neither you can acquire any knowledge about the inverses of the basic functions nor you can solve the advanced trigonometric problems.
Moreover, it is vital because the concepts related to trigonometry has a close relationship with calculus. Poor knowledge about trigonometry can restrict you to study higher mathematics in higher standards. The concepts of Cosine will be applicable in mathematical streams like differential calculus, integral calculus, and differential equations.
Solved Examples
Q1. What is the value of Cos θ in a right-angled triangle whose sides are 12cm and 20 cm respectively?
Ans. To find the value of cos θ in the triangle you need to apply the Pythagoras Theorem.
As per the Pythagoras theorem PR= sq. root [(12)2 + (20)2] = sq, root (544) cm.
12 / sq. root (544)
= 0.514
Q2. What are the laws of Cosines?
Ans: The laws of Cosines in terms of advanced trigonometry are dependent on the square root of the sides. You can find it on the Vedantu site.
FAQs on Cosine Meaning
1. What does cosine (cos) mean in simple terms?
In simple terms, cosine (cos) is one of the three primary trigonometric functions, alongside sine and tangent. For a given angle in a right-angled triangle, the cosine represents the ratio of the length of the adjacent side to the length of the hypotenuse. It essentially measures how much a corner 'points' in the direction of one of its arms.
2. How is the cosine of an angle calculated in a right-angled triangle?
To calculate the cosine of an angle (often denoted as θ) in a right-angled triangle, you use the following formula:
Cos(θ) = Length of the Adjacent Side / Length of the Hypotenuse
Here, the 'adjacent side' is the side next to the angle θ that is not the hypotenuse, and the 'hypotenuse' is the longest side, opposite the right angle. This formula is a fundamental concept in trigonometry as per the CBSE syllabus.
3. What are the values of cosine for standard angles (0°, 30°, 45°, 60°, 90°)?
The values of cosine for the most common angles are essential for solving trigonometric problems. As per the NCERT curriculum, these are:
- cos(0°) = 1
- cos(30°) = √3/2
- cos(45°) = 1/√2
- cos(60°) = 1/2
- cos(90°) = 0
Notice that as the angle increases from 0° to 90°, the value of its cosine decreases from 1 to 0.
4. What is the Law of Cosines and how does it differ from the Pythagorean theorem?
The Law of Cosines is a general formula that relates the lengths of the sides of any triangle to the cosine of one of its angles. For a triangle with sides a, b, and c, and the angle C opposite side c, the law is: c² = a² + b² - 2ab cos(C). The key difference is that the Pythagorean theorem (a² + b² = c²) only works for right-angled triangles, while the Law of Cosines can be applied to any triangle, regardless of its angles.
5. What is the fundamental difference between the sine and cosine functions?
The fundamental difference lies in which side of the right-angled triangle they relate to the hypotenuse. Cosine (cos) is the ratio of the adjacent side to the hypotenuse. In contrast, Sine (sin) is the ratio of the opposite side to the hypotenuse. Graphically, their wave functions are identical in shape but are out of phase by 90° (or π/2 radians); the cosine graph is essentially a sine graph shifted to the left.
6. Why is the function named 'cosine'? What is its relationship to 'sine'?
The name 'cosine' is short for 'complementary sine' or 'sine of the complementary angle'. In a right-angled triangle, the two non-right angles are complementary, meaning they add up to 90°. The cosine of one angle is always equal to the sine of its complementary angle. For instance, cos(30°) = sin(60°) and cos(70°) = sin(20°). This complementary relationship is the origin of its name.
7. How is the concept of cosine applied in real-world situations, such as in physics or engineering?
Cosine has numerous real-world applications. For example:
- In Physics, it's used to calculate the component of a force acting in a specific direction or to determine the work done by a force applied at an angle.
- In Engineering, it's used in architecture and construction to calculate lengths and angles for structures that are not right-angled.
- In Computer Graphics, it helps determine lighting and shading on 3D models based on the angle of a light source.
- In Navigation, it is used in GPS and astronomy to calculate distances and positions.
8. What does the graph of the cosine function look like and what are its key properties?
The graph of the cosine function, y = cos(x), is a continuous wave that oscillates between +1 and -1. Its key properties are:
- Maximum Value: The graph reaches its peak value of 1 at x = 0°, 360°, etc.
- Minimum Value: It reaches its minimum value of -1 at x = 180°, 540°, etc.
- Periodicity: The function is periodic with a period of 360° (or 2π radians), meaning its shape repeats every 360°.
- Symmetry: The cosine function is an even function, which means the graph is symmetric about the y-axis (cos(-x) = cos(x)).
9. What is a common mistake to avoid when using the Law of Cosines in a problem?
A common mistake is incorrectly handling the minus sign in the formula c² = a² + b² - 2ab cos(C). Students often forget that if the angle C is obtuse (greater than 90°), its cosine value, cos(C), will be negative. This results in a 'double negative' ( - 2ab × [a negative value] ), which effectively becomes an addition. Forgetting this can lead to a significantly smaller, incorrect value for the side 'c'.
10. How is the secant ratio related to the cosine ratio?
The secant (sec) ratio is the reciprocal of the cosine (cos) ratio. This means that for any given angle θ (where cos(θ) is not zero), the relationship is defined by the formula:
sec(θ) = 1 / cos(θ)
In a right-angled triangle, since cos(θ) = Adjacent/Hypotenuse, the secant is therefore sec(θ) = Hypotenuse / Adjacent.
