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A trigonometric equation or inequality contains a trigonometric ratio, where the unknown |(x)| is found in the argument.
A 2nd-degree trigonometric equation or inequality contains at least one squared trigonometric ratio or at least one product of 2 trigonometric ratios.
Since trigonometric functions are periodic functions, these types of equations may have no solution, one solution, several solutions, or an infinite number of solutions.
Also, we need to use angles in radians.
Sometimes, the trigonometric equations to be solved contain only one trigonometric ratio. Other times, they contain more than one ratio. The strategies for solving those that have more than one ratio vary.
Here are some strategies you can use when solving an equation or inequality involving more than one trigonometric ratio.
Use the definitions of the trigonometric ratios.
Use the trigonometric identities.
Rewrite the fractions using a common denominator.
Set the restrictions.
Perform a change of variables.
Use factoring or the quadratic formula.
Use the unit circle or the inverse functions |\arcsin,| |\arccos| and |\arctan| to determine the solutions.
Calculate the period in order to give all of the solutions.
Note: The first 3 strategies allow you to rewrite the whole equation only in terms of either sine or cosine, making the equation easier to solve. However, most of the time, you do not need to use all of these strategies, nor do you necessarily need to use them in this particular order.
Here's a reminder of the definitions and trigonometric identities often used to solve trigonometric equations.
||\begin{align}\tan(x)&=\dfrac{\sin(x)}{\cos(x)}\\[3pt]\text{cosec}(x)&=\dfrac{1}{\sin(x)}\\[3pt]\sec(x)&=\dfrac{1}{\cos(x)}\\[3pt]\cot(x)&=\dfrac{1}{\tan(x)}=\dfrac{\cos(x)}{\sin(x)}\end{align}||
||\begin{align}\cos(-x)&=\cos(x)\\[3pt] \sin(-x)&=-\sin(x) \end{align}||
||\begin{alignat}{13}&\cos^2(x)&&+\sin^2(x)&&=1\\[3pt]&\quad\,1&&+\tan^2(x)&&=\sec^2(x)\\[3pt] &\cot^2(x)&&+\quad1&&=\text{cosec}^2(x)\end{alignat}||
||\begin{align}\sin(A+B)&=\sin(A)\cos(B)+\cos(A)\sin(B)\\[2pt] \sin(A-B)&=\sin(A)\cos(B)-\cos(A)\sin(B)\\[2pt] \sin(2A)&=2\sin(A) \cos(A) \end{align}||
||\begin{align}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\[2pt] \cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\\[2pt] \cos(2A)&=\cos^2(A)-\sin^2 (A)\end{align}||
||\begin{align}\tan(A+B)&=\dfrac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}\\[5pt]\tan(A-B)&=\dfrac{\tan(A)-\tan(B)}{1+\tan(A)\tan(B)} \\[5pt] \tan(2A) &=\dfrac{2 \tan(A)}{1-\tan^2(A)} \end{align}||
When you use the inverse functions |\arcsin,| |\arccos| or |\arctan| on a calculator, you only get one angle value |(\theta).|
We find the other angle that has the same sine value as |\theta| by calculating |\pi - \theta.|
We find the other angle that has the same cosine value as |\theta| by calculating |- \theta.|
For the function |\arctan,| we don't need to find the other angle, since the period of a tangent function is |\pi| and not |2\pi.|
Given that |\cos(x) = \dfrac{2}{3}| and that |x| is an angle between |0| and |\dfrac{\pi}{2},| what is the value of the following expressions?
a) |\sec(x)|
b) |\sin(x)|
c) |\cot(x)|
d) |\tan(-x)|
Solve the following equation:||\tan^2(x)-3\sec(x)\tan(x)-\sec^2(x)=-1||
Solve the following equation:||2\sin^2(x)+\cos(2x)+1=0||
We also use the trigonometric identities of a sum or a difference to calculate the exact value of a trigonometric ratio.
Solve the following equation:||\sin(x)\cos(x)=2\cos(x)||
Solve the following equation:||15\sin(x)\cos(x)-2=5\sin(x)-6\cos(x)||
Solve the following equation:||3\tan(x)+\cot(x)=5\,\text{cosec}(x)||
To solve a trigonometric inequality, we use the same strategies as for solving a trigonometric equation. Then, once we have the solutions to the equation, we find the solution set of the inequation by testing values located on either side of the solutions found.
Also, always pay particular attention to the boundaries of the intervals of the solution set.
If the inequality sign is |<| or |>,| the boundaries are excluded.
If the inequality sign is |\leq| or |\geq,| the boundaries are included.
When a boundary corresponds to an asymptote (for the tangent function), it is always excluded.
Solve the following equation:||2\tan(x)+4\sec(x)\le 3\sin(x)+6||
Solve the following inequality: ||2\cos^2(x)<1-\sin(x)||