Subjects
Grades
Solving a Sine Equation or Inequality
Concept sheet | Mathematics
Secondary
5
Table of contents
- Top of page
- Solving a Sine Equation
- Solving a Sine Equation Using the Unit Circle
- Solving a Sine Equation Using |\boldsymbol{\arcsin}|
- Exercise - Solving a Sine Equation
- Solving a 2nd Degree Sine Equation
- Solving a Sine Inequality
- Solving a Sine Inequality Using the Unit Circle
- Solving a Sine Inequality Using |\boldsymbol{\arcsin}|
- Exercise - Solving a Sine Inequality
- Solving a 2nd Degree Sine Inequality
- See Also
A sine equation or inequality contains a sine ratio, where the unknown |(x)| is found in the argument.
Since the sine function is periodic, this type of equation may have no solution, one solution, several solutions, or an infinite number of solutions.
Also, we need to use the angles in radians.
In the unit circle, the sine of an angle corresponds to the y-coordinate of the point on the circle. When solving a sine equation, we look for angles that have a certain y-coordinate. To do so, we can use the main points of the unit circle or the inverse function |\boldsymbol{\arcsin}.|
When we use the unit circle, we generally choose angles located between |0| and |2\pi.|
When using the inverse function |\arcsin,| the result obtained is always an angle in the 1st quadrant or the 4th quadrant of the unit circle. In other words, the angle is between |-\dfrac{\pi}{2}| and |\dfrac{\pi}{2}.|
However, there are always 2 different trigonometric angles with the same y-coordinate. This is why, from the angle obtained |\boldsymbol{\color{#fa7921}{(\theta)}},| we find the 2nd angle by calculating |\boldsymbol{\color{#51b6c2}{\pi-\theta}}.|
The inverse function |\arcsin| is sometimes denoted |\sin^{-1},| especially on calculators.
Solving a Sine Equation
The procedure for solving a sine equation is as follows:
-
Isolate the sine ratio.
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Find the trigonometric angles.
- If the sine ratio is equal to the y-coordinate of one of the main points, use the unit circle.
- If not, use the inverse function |\boldsymbol{\arcsin}.| -
Solve the equations obtained with the trigonometric angles.
-
Calculate the period of the sine function.
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Give the solutions of the equation.
Solving a Sine Equation Using the Unit Circle
Here's an example using the main points of the unit circle to solve the equation.
Solve the following equation:||-\sin(x)-\dfrac{\sqrt{2}}{2}=0||
See solution
Solving a Sine Equation Using |\boldsymbol{\arcsin}|
Here's an example that uses the inverse function arcsine to solve the equation.
Solve the following equation for the interval |[0,2\pi].|||\sin\!\big(2(x-3)\big)-1=-\dfrac{1}{3}||
See solution
Exercise - Solving a Sine Equation
Solving a 2nd Degree Sine Equation
Here's an example of solving a 2nd degree sine equation.
Solve the following equation:||2\sin^2\left(\dfrac{x+\pi}{3}\right)-\sin\left(\dfrac{x+\pi}{3}\right)-1=0||
See solution
Solving a Sine Inequality
The procedure for solving a sine inequality is as follows:
-
Change the inequality symbol to an equal symbol.
-
Isolate the sine ratio.
-
Determine the trigonometric angles.
- If the sine ratio is equal to a y-coordinate of one of the main points, use the unit circle.
- If not, use the inverse function |\boldsymbol{\arcsin}.| -
Solve the equations obtained with the trigonometric angles.
-
Calculate the period of the sine function.
-
Give the solution set of the inequality.
Solving a Sine Inequality Using the Unit Circle
Here's an example using the main points of the unit circle to solve the inequality.
Solve the following inequality:||4\sin(4x)+3>5||
See solution
Solving a Sine Inequality Using |\boldsymbol{\arcsin}|
Here's an example of how to use the inverse function arcsine to solve the inequality.
Solve the following inequality:||-10\sin\left(\dfrac{2(x-1)}{3}\right)+4\ge12||
See solution
Exercise - Solving a Sine Inequality
Solving a 2nd Degree Sine Inequality
Here's an example of solving a 2nd degree sine inequality.
Solve the following inequality:||\sin^2(x)<\dfrac{3}{4}||