Sin Theta

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,

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

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 \(cosec(\theta)\).

Graph of the sin theta function
Graph of the sin theta function

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

RadiansDegreeTangent Value

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, 

\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}\\

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.


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.

More Trigonometric Functions

Cos ThetaTan Theta
Sec ThetaCot Theta
Cosec Theta

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