Graphs of Trigonometric Functions

Find the solutions to $\sin \left(x\right)=0.9$, for the interval $0°\le x\le 360°$.

Solution:

The first step is to sketch the graph of $y=\sin \left(x\right)$ and $y=0.9$ on the same axis for the interval $0°\le x\le 360°$.

Graphs of Trigonometric Functions- Graph showing solutions of sin(x)=0.9, Jordan Madge- StudySmarter Originals

The points of intersection have been labelled in orange as 1 and 2, these are the solutions we are seeking to find the exact values of.

The second step is to find the exact value of the initial solution. This can be done by typing ${\sin }^{-1}(0.9)$ into our calculator. When we do this, we obtain $x=64.2°$. This is clearly the first solution labelled on the diagram since it is between $0°$and $90°$.

It is important to note that your calculator should be in degree mode when calculating trigonometric values since we are working in degrees. If your calculator is in radian mode, the answer may vary and so you may obtain the incorrect answer. You know that your calculator is in degree mode when a small D appears on the top of the screen. If you see an R or any other letter, it is in the wrong mode and needs to be changed.

The next step is to find the other solution using the symmetrical property of the graph of sin(x). If we notice, the graph is symmetrical about id="2569857" role="math" $x=90°$. Thus, we can work out the second solution by working out the distance between $64.2°$ and $90°$, and then adding this value to $90°$. This can be illustrated in the below diagram:

Graphs of Trigonometric Functions- Graph showing solutions of sin(x)=0.9, Jordan Madge- StudySmarter Originals

Since the distance between $64.2°$ and $90°$ is $25.8°$, the second solution is id="2569881" role="math" $90+25.8=115.8°$. Therefore, the two solutions to the equation $\sin \left(x\right)=0.9$ in the interval $0°\le x\le 360°$ are id="2569884" role="math" $x=64.2°$ and id="2569885" role="math" $x=115.8°$.

Find the solutions to $\cos \left(x\right)=-0.2$ for the interval $-180°\le x\le 180°$.

Solution:

The first step is to sketch the graphs of $y=\cos \left(x\right)$ and id="2569887" role="math" $y=-0.2$ on the same axes for the interval $-180°\le x\le 180°$ so that we can see the solutions we are trying to find.

Graphs of Trigonometric Functions- Graph showing solutions of cos(x)=-0.2, Jordan Madge- StudySmarter Originals

The next step is to find the initial solution by typing ${\cos }^{-1}(-0.2)$ into our calculator. We obtain id="2569892" role="math" $x=101.5°$. Clearly, this is the solution labelled 2 on the diagram, since it is a little more than id="2569898" role="math" $90°$ but less than id="2569899" role="math" $180°$.

We now need to find the other solution depicted in the diagram. Since the graph of $\cos \left(x\right)$ is symmetrical about the line $x=0$, we can see that the other solution must be at id="2569894" role="math" $x=-101.5°$. Thus, the two solutions to id="2569895" role="math" $\cos \left(x\right)=-0.2$ in the interval $-180°\le x\le 180°$ are id="2569897" role="math" $x=101.5°$ and id="2569896" role="math" $x=-101.5°$.

Find the solutions to $\tan \left(x\right)=2.3$ for the interval $0°\le x\le 360°$.

Solution:

The first step, as usual, is to sketch the graphs $y=\tan \left(x\right)$ and id="2569908" role="math" $y=2.3$ on the same axes for the interval $0°\le x\le 360°$.

Graphs of Trigonometric Functions- Graph showing solutions of tan(x)=2.3, Jordan Madge- StudySmarter Originals

We can see that there are two points of intersection and thus two solutions to $\tan \left(x\right)=2.3$. The first solution can be found by typing id="2569911" role="math" ${\tan }^{-1}(2.3)$ into our calculator. Doing so, we obtain id="2569912" role="math" $x=66.5°$This is clearly the first solution since it is between $0$ and $90$ degrees.

The graph of $\tan \left(x\right)$ repeats itself periodically after $180$ degrees. Therefore, we can find the next solution by adding multiplies 180 to the initial solution. So, the second solution is at id="2569918" role="math" $180+66.5=246.5°$. Therefore, the two solutions to $\tan \left(x\right)=2.3$ in the interval $0°\le x\le 360°$ are id="2569915" role="math" $x=66.5°$ and id="2569917" role="math" $x=246.5°$.

Solutions to any equations involving tan(x) can be found by adding multiples of 180 to the initial solution.

Find the solutions of $4\tan \left(x\right)=3$ for the interval $-180°\le x\le 180°$.

Solution:

We cannot solve this equation in its current form. We first need to divide both sides by $4$ to get $\tan \left(x\right)$ by itself. We obtain $\tan \left(x\right)=\frac{3}{4}$. Now, we can find the first solution to the equation by taking the inverse tan of both sides to get $x={\tan }^{-1}\left(\frac{3}{4}\right)=36.9°$.

Now, since it is tan, we know that solutions can be found by adding or subtracting multiples of $180°$ to the initial solution. Thus, the next solution will be at $36.9+180=216.9$, however, this is out of the range. We can get another solution by subtracting$180°$ from $36.9°$ to get $-143.1°$ which is in the range. Subtracting a further ${180}^{°}$ will yield a solution out of the range, therefore the two solutions to $4\tan \left(x\right)=3$ in the interval $-180°\le x\le 180°$ are $x=36.9°$ and $x=-143.1°$.