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
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).
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.
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.
More Trigonometric Functions
