Value of Cos 180°

Trigonometric functions (also known as circular functions, angle functions, or goniometric functions) are real functions that connect the angle of a right-angled triangle to ratios of two side lengths in mathematics. Trigonometry is the study of triangle measurements, concentrating on the triangle's length, height, and angles. They are widely utilised in all geodetic studies, including navigation, solid mechanics, celestial mechanics, geodesy, and many more. They are among the simplest periodic functions, and as a result, they are frequently employed in Fourier analysis to analyse periodic events.

In the actual world, trigonometric functions are quite useful. Engineering, architecture, satellite navigation, medical imaging, and sound wave creation are just a few examples. Engineers, architects, and astronauts all utilise trigonometry functions to compute various measures that are affected by triangle characteristics. It generates sound and light waves using the wave pattern of trigonometric functions in various applications. The sine and cosine wave patterns are used in the development of computer music.

In this article we are going to discuss cos angle values, how to derive them, fun facts related to trigonometry ratios, solved problems, and also a few of the most important questions related to cos angle will also be answered.

Cos Angle Formula

There are numerous formulae in trigonometry, but when it comes to a right-angle triangle, there are just a few key basic formulas. The ratio of the neighbouring side to the hypotenuse, which is one of the acute angles, is known as the Cos theta or cos. The following is the cosine formula:

Cos θ = $\left(\frac{Adjacent ~ side}{Hypotenuse~Side}\right)$

The above image shows the right-angle triangle.

Value of cos 1800

The value of Cos 180° (π) is equal to -1

How to obtain the value of Cos 180°?

The value of cos 180 degrees, or cos pi, may be expressed in terms of several angles such as 0°, 90°, and 270°.

Consider the unit circle, which in the Cartesian plane is divided into four quadrants. The value of cos 180 degrees from the Cartesian plane is calculated using the value 180 degrees in the second quadrant. Because the second quadrant's cosine values are always negative.

From the cos value equal to 0, we will derive the value of cos 180°.

The exact value of cos 0 degrees is 1 as we know.

As a result, the cos 180 degree is -(cos 0), which equals - (1)

Hence, cos 180 degrees is equal to -1.

Radians are also used to represent it.

So, the value of cos = -1, when the value of = 180.

There are a few more ways for cosine derivation. To determine the value, several degree values of sine and cosine functions are obtained from the trigonometry table.

As we know,

180° – 0° = 180° ——— (A)

270° – 90° = 180° ——— (B)

Method A: Cosine 180-degree Derivation

Now use the above representation in cosine function relations, then we get,

The cos pi value =Cos 180° = cos(180° – 0°) = – cos 0° = – (1) = – 1

Hence, the cos pi value = -1

Method B: Cosine π Value Derivation

According to the above B equation,

The value of Cos 180°= cos(270° – 90°)

Cos 180°= – sin 90° (We know that the given relationship with sin, cos ( 270° – a ) = – sin a)

The Sin 90 degrees has a value of 1.

When we replace the value in the above equation, we obtain

The value of Cos 180° = – (1) = -1

Hence, the value of cos 180° is -1.

The values of the trigonometric ratios for various angles are given below:

a) Cos 180 Minus x

i.e cos (1800 - x) = -cos x

Explanation:

One of the simplest ways could be to use the formula

⇒ cos(A-B) = cosA cosB +sinAsinB

Hence cos(1800-x) = cos 1800 cosx + sin 1800sinx

⇒ (-1)cosx + 0$\times$sinx

⇒ -cosx

Hence it’s solved

b) Cos 180 Plus x

cos (1800+x) = -cosx

Use the summation method in cosine identity,

⇒ cos(A+B) = cosA cosB - sinA sin B

⇒ cos(1800)cosx- sin(1800)sinx = -cosx

⇒ -1(cosx) - 0(sinx) = -cosx

⇒ -cosx = -cosx

⇒ LHS = RHS

Hence it’s solved.

Fun Fact

Real-Life Uses of Trigonometric Functions:

• It's used in oceanography to figure out how high the tides are in the seas.

• The sine and cosine functions are essential in the study of periodic functions, which include sound and light waves.

• Trigonometry and Algebra make up Calculus.

• Trigonometry may be used to roof a home, make the roof slanted (in the case of single-family bungalows), and calculate the height of a building's roof, among other things.

• It's often used in the marine and aviation sectors.

• It’s used to form cartography (creation of maps).

• Satellite navigation systems also use trigonometry.

• Many real-life events may be represented using sine and cosine functions, such as radio waves, tides, musical tones, and electrical currents.

Exemplar Problems with Step by Step Solution:

1 .If a = 18, b = 24, and c = 30, then find the value of Cos A

Ans: By definition, we know that in the right-angled triangle with a length of the side opposite to the angle θ as perpendicular (P), the base (B) and hypotenuse (H):

Sinθ =$\left(\frac{{P}}{H}\right)$, Cosθ =$\left(\frac{{B}}{H}\right)$, tanθ=$\left(\frac{{P}}{B}\right)$,

It is given that the side of the triangle is a = 18, b = 24 and c = 30,

Now using the trigonometric ratios we have the value of cos θ as,

Cosθ =$\left(\frac{{B}}{H}\right)=\left(\frac{{24}}{30}\right)=\left(\frac{{4}}{5}\right)$

2. If Sin B = ½ and CosC = ½ then B+C =?

Ans: In the question, it is given that Sin B = ½ which is equal to Sin 300

⇒sin B = sin 300

⇒B = 300

Similarly,

⇒cos C = ½

⇒cos C = cos 600

⇒C = 600

Now B+C = 600+300=900

Multiple choice questions:

1. The value of cos (90° – θ) and sin θ

1. Not the same

2. Same

3. No relation

4. Incorrect question

Ans: From the trigonometric identities we know that cos(90° – θ) = sin θ

Hence the correct answer is option (B)

2. What would be the value of sin X if cos X = = a/b

Ans: From the question cos X = $\left(\frac{{a}}{b}\right)$

By using the trigonometric identities we have,

sin2X+cos2X= 1

sin2X= 1-cos2X= 1- $\left(\frac{a}{b}\right)^{2}$

sinX=$\left(\frac{\sqrt{b^2-a^2}}{b}\right)$

Conclusion:

Throughout the article, we have discussed the cos angle values, their definition, fun facts and also a few exemplar problems related to trigonometry. Students can visit the official website of Vedantu where we have enough information regarding trigonometry and some of the most important problems. They can practice these questions and excel in this topic.