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Value of Cos 120 is -½. Cos 120 degrees is equivalent to minus half, and is often written as (- ½ ) or (-0.5). Cos, in trigonometric expressions, is identified as one of the main functions that often conforms with the relationship between the angles and sides present in a right-angled triangle.
Thus, in a right-angled triangle, there are:
- Hypotenuse: The longest side of the triangle that can be found opposite to 90 degrees is the Hypotenuse.
- Perpendicular (Opposite): Perpendicular is known as the side which is opposite to the unknown angle and perpendicular to the base (which means, the angle between the base and the perpendicular is 90 degrees).
- Base (Adjacent): The Adjacent is the side where the triangle stands, and contains both angles (which are the 90 degrees and unknown angle).
Also read: Isosceles Triangle Theorems
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Key Terms: Right-Angle Triangle, Cos 120, Sine, Cosine, Tan, Hypotenuse, Angle, Trigonometric Ratios, Value Table, Adjacent side, Opposite Side, Right Angle Triangle
What is a Right-Angled Triangle?
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A triangle is a closed figure that has three sides and three angles connected with one another. A right-angled triangle often forms when one of the interior angles is 90 degrees, while the remaining two are less than 90 degrees respectively.
- A right-angled triangle is considered to be a type of triangle which has one of the angles that is equivalent to 90 degrees. The other two angles, however, are known to sum up to 90 degrees.
- The sides including the right angle are perpendicular and the base of the triangle respectively.
- The third and longest side is called the hypotenuse.
The three sides of the right triangle are known to be related to one another. This relationship they have can also be illustrated using the Pythagoras theorem. According to it, the theorem used is:
→ Hypotenuse2 = Perpendicular2 + Base2 Solved ExampleQues. For a right triangle, consider that the perpendicular = 8 cm and the base = 6 cm. Thus, determine the value of the hypotenuse. Ans. For the given question, Perpendicular = 8 cm Base = 6cm Thus, we need to determine the Hypotenuse. Hence, by applying Pythagoras theorem, we can see that: Hypotenuse = √(Perpendicular2 + Base2) H = √(62 + 82) = √36 + 64 = √100 = 10 cm Hence, the hypotenuse of the right triangle is 10 cm. |
Trigonometric Functions Detailed Video Explanation
Value of Cos 120
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Cos 120, as per trigonometric identities, can either be denoted with - ½ or -0.5 with respect to how it is supposed to be used in the derivation value of cos 120. However, Cos 120 degrees, if considered in radians, is often identified as cos (120° × π/180°), which is, cos (2π/3) or cos (2.094395. . .).
Prior to understanding the methods related to finding the value of Cos 120, it is essential to know the various expressions it can be denoted with.
- Cos 120°: -0.5
- Cos 120°, as per fraction: -(1/2)
- Cos (-120 degrees): -0.5
- Cos 120°, as per radians: cos (2π/3) or cos (2.0943951 . . .)
If using the degree-to-radian conversion in the Derivation value of cos 120, then:
θ (radians) = θ in degrees × (pi/180°)
⇒ 120 degrees = 120° × (π/180°) rad
= 2π/3 or 2.0943 . . . (where pi = 22/7 or 3.14, as per requirement)
Therefore, cos 120° = cos(2.0943) = -(1/2) or -0.5.
Cos 120, or other simple trigonometric functions relevant to the Derivation value of cos 120, are put into use for several applications. For instance, it helps in forming computer music since sounds especially travel via waves. Cos 120 is the additive inverse of values of cosine 60 degrees or cos 60 degrees. It can be identified as a ratio of the base of a right-angled triangle to its respective hypotenuse.
Also Read:
| Related Articles | ||
|---|---|---|
| Sine Squared X | Sin 30 Degrees | Sin 90 Degrees |
| Sin 180 Degrees | Cos 120 Degrees | Sin 30 Degrees |
Finding Value of Cos 1200
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The angles of an individual right-angled triangle can be implied in several forms of multiples or sub-multiples of 180°, or pi in the case of radians.
Simplifying the same, the cosine function is usually negative in the 2nd quadrant, out of the four quadrants. Since the value of cos 120° is identified as -0.5, its expression can be implied by using two separate methods,
- Unit Circle
- Trigonometric Functions
120° can be illustrated in terms of two angles, which are, either 90° or 180°.
Thus, it can be considered from the angles 90 degrees and 180 degrees.
180° – 60° = 120° ... (1)
90° + 30° = 120° ... (2)
After applying the same, we can see:
Cos 120° = cos(180° – 60°) = – cos 60° = -½ (because cos(180° – x) = – cos x)
Cos 120° = cos(90° + 30°) = – sin 30° = -½ (because cos (90° + x) = -sin x)
Trigonometry Table
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Other trigonometric identities can be represented in the following Trigonometry Table:
| Type | Values | |||||||
|---|---|---|---|---|---|---|---|---|
| Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| SinΘ | 0 | 1/2 | 1√212 | √3232 | 1 | 0 | -1 | 0 |
| CosΘ | 1 | √3232 | 1√212 | 1/2 | 0 | -1 | 0 | 1 |
| TanΘ | 0 | 1√313 | 1 | √33 | Not Defined | 0 | Not Defined | 0 |
| CosecΘ | Not Defined | 2 | √22 | 2√323 | 1 | Not Defined | -1 | Not Defined |
| SecΘ | 1 | 2√323 | √22 | 2 | Not Defined | -1 | Not Defined | 1 |
| CotΘ | Not Defined | √33 | 1 | 1√313 | 0 | Not Defined | 0 | Not Defined |
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Things to Remember
- The value of Cos 120 is 0.5 or ½.
- The Pythagoras Theorem is, Hypotenuse2 = Base2 + Perpendicular2
- In trigonometric identities, Cos is one of the main functions that form a relationship between the angles and sides present in a right-angled triangle.
- The cosine function is known to be negative in the 2nd quadrant, out of the other four quadrants.
-
Because the value of cos 120° is implied as -0.5, its expression can also be represented by two methods, namely, Unit Circle and Trigonometric Functions.
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Sample Questions
Ques. Determine the value of cos 120 degrees using the unit circle. (4 marks)
Ans. Consider P (a,b) to be a point on the circle forming an angle AOP = x radian. This clearly means that the length of the arc AP is equivalent to x. Therefore, cos x = a, while sin x = b.
Hence, we can determine the value of Cos 120° from the value of 60°,
It is clear that:
Cos 60 = 12
Hence, we are also aware that
cos (180-x) = - cos x
Thus,
Cos 120° =-cos 60 = - 12
That means,
Cos 120° = -12
Ques. What is the value of Cos 120 degrees? (2 marks)
Ans. Cos 120°= cos (90 degree + 30 degree)
= - sin 30°
= - ½
Hence, the answer is - ½.
Ques. Determine the value of (cos2 60° - sin2 60°) by using Cos 120° as - 0.5. (2 marks)
Ans. When considering the Cos 2a formula, we can further apply,
(cos2 60° - sin2 60°) = cos (2 × 60°) = cos 120°
Therefore, cos 120° = - 0.5
⇒ (cos2 60° - sin2 60°) = - 0.5
The answer thus is - 0.5
Ques. Considering the value cos 120°, find the following expression, (1 - sin2 (120°)). (2 marks)
Ans. Since we are already aware that
(1-sin2(120°)) = (cos2(120°)) = 0.25
⇒ (1-sin2(120°)) = 0.25
Ques. Determine the value of 2 cos(120°)/3 sin(-30°), by using the required formula of cos 120 degrees. (2 marks)
Ans. If we put trigonometric identities into use, then it is quite clear that cos (120°) = Sin (90° - 120°) = Sin ( - 30°)
cos(120°) = sin(-30°)
Therefore, Value of 2cos(120°)/3 sin(-30°) = 2/3
Ques. Determine the various forms in which Cos 120 degrees can be signified. (2 marks)
Ans. Cos 120 degrees can be denoted in several forms, such as,
- Cos 120°: -0.5
- Cos 120°, as per fraction: -(1/2)
- Cos (-120 degrees): -0.5
- Cos 120°, as per radians: cos (2π/3) or cos (2.0943951 . . .)
Ques. Can you determine the appropriate value of Cos 120 Degrees with respect to Tan 120°? (2 marks)
Ans. Since we are aware that cos 120 degrees can be signified, with respect to trigonometric expressions, as -1/√(1 + tan2 (120°)).
Thus, it can be said that the value of Tan 120 degrees can be considered as -1.732050.
Ques. How can you determine the value of cos 120 degrees with respect to other trigonometric identities? (2 marks)
Ans. By using the trigonometric identities, the value of cos 120 degrees can be portrayed in form of different trigonometric expressions, such as
- ± √(1-sin2(120°))
- ± 1/√(1 + tan2(120°))
- ± cot 120°/√(1 + cot2(120°))
- ± √(cosec2(120°) - 1)/cosec 120°
- 1/sec 120°
Ques. Consider that the hypotenuse is given as 13 cm, along with the base as 12 cm. Thus, determine the length of the perpendicular of the right triangle. (3 marks)
Ans. As per the given question,
Hypotenuse = 13 cm
Base = 12 cm
Thus, we need to determine the Perpendicular.
By the help of Pythagoras' theorem, we are aware that,
Hypotenuse2 = Perpendicular2 + Base2
Perpendicular2 = Hypotenuse2 – Base2
P = √(132 – 122)
= √(169 – 144)
= √25
= 5 cm
Hence, the value of the perpendicular is 5 cm.
Ques. Simplify the following: 4 (cos 210°/sin 300°). (2 marks)
Ans. We are aware that:
cos 210° = sin 300°
Thus,
⇒ 4 cos 210°/sin 300° = 4 (cos 210°/cos 210°)
= 4(1) = 4
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