Cosine Function

Cos function (or cosine function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine(co+sine). There are various topics that are included in the entire cos concept. Here, the main topics that are focussed include:

  • Cosine Definition
  • Cosine Formula
  • Cosine Table
  • Cosine Properties With Respect to the Quadrants
  • Cos Graph
  • Inverse Cosine (arccos)
  • Cosine Identities
  • Cos Calculus
  • Law of Cosines in Trigonometry
  • Additional Cos Values
  • Cosine Worksheet
  • Trigonometry Related Articles for Class 10
  • Trigonometry Related Articles for Class 11 and 12
  • Other Trigonometry Related Topics

Other Trigonometric Functions

Sine Function

Tan Function

Cosec (Csc) Function

Sec Function

Cot Function

Cosine Definition

In a right-triangle, cos is defined as the ratio of the length of the adjacent side to that of the longest side i.e. the hypotenuse. Suppose a triangle ABC is taken with AB as the hypotenuse and \(\alpha\) as the angle between hypotenuse and base.

Cosine Function

Now, for this triangle,

\(cos\, \alpha\, =\, \frac{Adjacent\,Side}{Hypotenuse}\)

Cosine Formula

From the definition of cos, it is now known that it is the adjacent side devided by the hypotenuse. Now, from the above diagram,

\(cos\, \alpha\, =\, \frac{AC}{AB}\)

Or,

\(cos\, \alpha\, =\, \frac{b}{h}\)

Cosine Table

Cosine Degrees

Values

Cos 0°

1

Cos 30°

√3/2

Cos 45°

1/√2

Cos 60°

1/2

Cos 90°

0

Cos 120°

-1/2

Cos 150°

-√3/2

Cos 180°

-1

Cos 270°

0

Cos 360°

1

Cosine Properties With Respect to the Quadrants

It is interesting to note that the value of cos changes according to the quadrants. In the above table, it can be seen that cos 120, 150 and 180 degrees have negative values while cos 0, 30, etc. have positive values. For cos, the value will be positive in the first and the fourth quadrant.

Cosine Function

Degree Range

Quadrant

Cos Function Sign

Cos Value Range

0 to 90 Degrees

1st Quadrant

+ (Positive)

0 < cos(x) < 1

90 to 180 Degrees

2nd Quadrant

– (Negative)

-1 < cos(x) < 0

180 to 270 Degrees

3rd Quadrant

– (Negative)

-1 < cos(x) < 0

270 to 360 Degrees

4th Quadrant

+ (Positive)

0 < cos(x) <10

Cos Graph

The cosine graph or the cos graph is an up-down graph just like the sine graph. The only difference between sine graph and cos graph is that sine graph starts from 0 while the cos graph starts from 90 (or \(\frac{\pi}{2}\)). The cos graph given below starts from 1 and falls till -1 and then starts rising again.

Cosine Function

Arccos (Inverse Cosine)

The cos inverse function can be used to measure the angle of any right angled triangle if the ratio of the the adjacent side and hypotenuse is given. The inverse of sine is denoted as arccos or \(cos^{-1}\).

For an right triangle with sides 1, 2, and \(\sqrt{3}\), the cos function can be used to measure the angle.

Cosine Function

In this, the cos of angle A will be, cos(a)= adjacent/hypotenuse.

So, \(cos(a)\,=\,\frac{\sqrt{3}}{2}\)

Now, the angle “a” will be \(cos^{-1}(\frac{\sqrt{3}}{2})\)

Or, \(a\,=\,\frac{\pi}{6}\,=\,30\)°

Important Cos Identities

  • \(cos^{2}(x)\,+\,sin^{2}(x)\,=\,1\)
  • \(cos\,\theta\,=\,\frac{1}{sec\,\theta}\)
  • \(cos(-\theta)\,=\,cos(\theta)\)
  • \(arccos(cos(x))\,=\,x\,+\,2k\pi\; where \,k=integer\)
  • \(Cos(2x)=cos^{2}(x)\,-\,sin^{2}(x)\)
  • \(cos(\theta)\,=\,sin(\frac{\pi}{2}\,-\,\theta)\)

Below, all the other trigonometric functions in terms of cos function are also give.

Other Trigonometric Functions in Terms of Sine

Trigonometric Functions

Represented as Sine

\(sin\,\theta\) \(\pm \sqrt{1-cos^{2}\theta}\)
\(tan(\theta)\) \(\pm \frac{\sqrt{1-cos^{2}\,\theta}}{cos\,\theta}\)
\(cot(\theta)\) \(\pm\frac{cos\,\theta}{\sqrt{1\,-\,cos^{2}\,\theta}}\)
\(sec(\theta)\) \(\pm\frac{1}{cos\,\theta}\)
\(cosec(\theta)\) \(\pm\frac{1}{\sqrt{1\,-\,cos^{2}\,\theta}}\)

Cos Calculus

For cosine function \(f(x)\,=\,cos(x))\), the derivative and the integral will be given as:

  • Derivative of cos(x), \(f'(x)\,=\,-sin(x))\)
  • Integral of cos(x), \(\int f(x)\,dx\,=\,-sin(x)\,+\,C)\) (where C is the constant of integration)

Law of Cosines in Trigonometry

The law of cosine or cosine rule in trigonometry is an relation between the side and the angles of a triangle. Suppose a triangle with sides a, b, c and with angles A, B, C are taken, the cosine rule will be as follows.

According to cos law, the side “c” will be:

\(c^{2}\,=\,a^{2}\,+\,b^{2}\,-\,2ab\,cos(C)\)

It is important to be thorough with the law of cosines as questions related to it are common in the examinations.

Also Check:

  • Law of Sines
  • Tan Law

Additional Cos Values

Cos 1 Degree is 0.99

Cos 2 Degree is 0.99

Cos 5 Degree is 0.996

Cos 8 Degree is 0.990

Cos 10 Degree is 0.984

Cos 15 Degree is 0.965

Cos 20 Degree is 0.939

Cos 30 Degree is 0.866

Cos 40 Degree is 0.766

Cos 50 Degree is 0.642

Cos 70 Degree is 0.342

Cos 80 Degree is 0.173

Cos 100 Degree is -0.173

Cos 105 Degree is -0.258

Cos 210 Degree is -0.866

Cos 240 Degree is -0.5

Cos 270 Degree is 0

Cos 330 Degree is 0.866

Cos Questions (Worksheets)

  1. \(sin(cos^{-1}\frac{3}{5})\)
  2. In a triangle PQR, PR is 14 cm, QR is 10 cm, and angle RQP is 63 degrees. Calculate angle RPQ and the length of PQ.
  3. In triangle ABC, AB 6 cm, AC is 13 cm, and angle CAB is 91 degrees. Calculate the length of BC.
  4. Derive the value of cos 60 geometrically.
  5. A ramp is pulled out of the back of a truck. There is a 38 degrees angle between the ramp and the pavement. The distance from the end of the ramp to to the back of the truck is 10 feet. Calculate the length of the ramp?

Trigonometry Related Articles for Class 10

Trigonometry Related Articles for Class 11 and 12

Other Trigonometry Related Topics

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Practise This Question

What will you substitute to make the following equation a linear equation

1x + 3y = 1