The cosine function ‘or’ **Cos Theta** is one of the three most common trigonometric functions along with sine and tangent. In right-angled trigonometry, the cosine function is defined as the ratio of the adjacent side and hypotenuse.

The mathematical denotation of the sine function is,

\(\cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse Side}}\)Index

## More About Cos Theta

Cos theta formula can also be calculated from the product of the tangent of the angle with the sine of the angle.

\(\cos(\theta) = \frac{\sin(\theta)}{ \tan(\theta)}\)The derivative of \(\cos(\theta)\) in calculus is \(-\sin(\theta)\) and the integral of it is \(\sin(\theta)\). The reciprocal of cos theta is sec theta.

Below is a table of cos theta values for different degrees and radians.

Radians | Degree | Tangent Value |

0 | 0° | 0 |

\(\frac{\pi}{6}\) | 30° | \(\frac{\sqrt{3}}{2}\) |

\(\frac{\pi}{4}\) | 45° | \(\frac{1}{\sqrt{2}}\) |

\(\frac{\pi}{3}\) | 60° | \(\frac{1}{2}\) |

\(\frac{\pi}{2}\) | 90° | 0 |

\(\pi\) | 180° | -1 |

\(\frac{3\pi}{2}\) | 270° | 0 |

\(2\pi\) | 360° | 1 |

## Important Cos Theta Formula

Some important properties of the cosine function and cos theta formula are:

- \(\cos(-x) = -\cos(x)\)

- \(\cos(90° – x) = \sin(x)\)

- \(\cos(x + 2\pi) = \cos(x)\)

- \(\cos(\pi – x) = -\cos(x)\)

- \(\cos^2(x) + \sin^2(x) = 1\)

- \(cos(x + y) = \cos(x)*\cos(y) + \sin(x)*\sin(y)\)

- \(\cos(x – y) = \cos(x)*\cos(y) + \sin(x)\sin(y)\)

- \(\cos(2x) = \cos^2(x) – \sin^2(x)\)

- \(\cos(3x) = 4\cos^3(x) – 3\cos(x)\)

- \(\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos(x)}{2}}\)

- \(\cos(x) + \cos(y) = 2 \cos(\frac{x + y}{2}) \cos(\frac{x – y}{2})\)

- \(\cos(x) – \cos(y) = -2 \sin(\frac{x + y}{2}) \sin(\frac{x – y}{2})\)

- \(\cos(x) = \frac{1 – \tan^2(\frac{x}{2})}{1 + \tan^2(\frac{x}{2})}\)

**Solved Examples**

**Question 1. If \(\sin(x) = \frac{6}{13}\), calculate the value of \(\cos(x)\).**

**Solution.** Using trigonometric identity,

\(\cos^2(x) = 1 – \sin^2(x) = 1 – \frac{36}{169}\)

\(\cos^2(x) = \frac{133}{169}\)

\(∴ \cos(x) = \frac{\sqrt{133}}{13}\)

**Question 2. If \(\tan(\frac{x}{2}) = \frac{5}{8}\), calculate the value of \(\cos(x)\).**

**Solution.** Using trigonometric identity,

\begin{align}

\cos(x) & = \frac{1 – \tan^2(\frac{x}{2})}{1 + \tan^2(\frac{x}{2})}\\

& = \frac{1 – (\frac{5}{8})^2}{1 + (\frac{5}{8})^2}\\

& = \frac{39/64}{89/64}\\

\cos(x) & = \frac{80}{89}\\

\end{align}

\)

**Question 3. **Consider a ladder leaning against a brick wall making an angle of 60^{o} with the horizontal. If the ground distance between the ladder and the wall is 10 ft, then up to what height of the wall the ladder reaches?

**Solution.** Given, Angle between ladder and ground, \(\theta = 60\) and distance from ladder to wall = 10 ft.

As we know,

\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)

\(\cos(60°) = \frac{\text{10}}{Hypotenuse}\)

\(\frac{1}{2} = \frac{\text{10}}{Hypotenuse}\)

\(\text{Hypotenuse} = 20\)

∴ The height of the wall up to which the ladder reaches is 20 ft.

**FAQs**

**Explain how cos(-x) = cos(x).**As we know, the angle (-x) lies in the 4th quadrant of a graph, and cosine is positive in this quadrant. Hence, this shows that cos(-x) = cos(x).

**What is cos theta****?**Sin theta of a right-angled triangle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse.

**In which quadrants is the cosine function positive and in which quadrants is it negative?**

It can be observed from the above graph that cosine function is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd quadrants.

More Trigonometric Functions