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

The mathematical denotation of the sine function is,

\(\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse Side}}\)Index

## More About Sin Theta

Sin theta formula can also be calculated from the product of the tangent of the angle with the cosine of the angle.

\(\sin(\theta) = \tan(\theta) \times \cos(\theta)\)

The derivative of \(\sin(\theta)\) in calculus is \(\cos(\theta)\) and the integral of it is \(-\cos(\theta)\). The reciprocal of sin theta is \(cosec(\theta)\).

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

Radians | Degree | Tangent Value |

0 | 0° | 0 |

\(\frac{\pi}{6}\) | 30° | \(\frac{1}{2}\) |

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

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

\(\frac{\pi}{2}\) | 90° | 1 |

\(\pi\) | 180° | 0 |

\(\frac{3\pi}{2}\) | 270° | -1 |

\(2\pi\) | 360° | 0 |

## Important Sin Theta Formula

Some important properties of the sine function and sin theta formula are:

- \(\sin(-x) = -\sin(x)\)

- \(\sin(90° – x) = \cos(x)\)

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

- \(\sin(\pi – x) = \sin(x)\)

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

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

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

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

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

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

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

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

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

**Solved Examples**

**Question 1. If \(\cos(x) = \frac{3}{5}\), calculate the value of \(\sin(x)\).**

**Solution.** Using trigonometric identity,

\(\sin^2(x) = 1 – \cos^2(x) = 1 – \frac{9}{25}\)

\(\sin^2(x) = \frac{16}{25}\)

\(∴ \sin(x) = \frac{4}{5}\)

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

**Solution.** Using trigonometric identity,

\begin{align}

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

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

& = \frac{5/4}{89/64}\\

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

\end{align}

\)

**Question 3. John was working on a construction site. He wants to reach the top of the wall. A 44 ft long ladder connects a point on the ground to the top of the wall. The ladder makes an angle of 60 degrees with the ground. What would be the height of the wall?**

**Solution.** Given, Angle between ladder and ground, \(\theta = 60\) and Hypotenuse = 44 ft.

As we know,

\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)

\(\sin(60°) = \frac{\text{Opposite}}{44}\)

\(\frac{\sqrt{3}}{2} = \frac{\text{Opposite}}{44}\)

\(\text{Opposite} = 22 \sqrt{3}\)

∴ The height of the wall is \(22 \sqrt{3}\) feet.

**FAQs**

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

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

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

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

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