solving trig equations
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Solve the equation for cos(x) first.
In other words, you want to isolate the expression cos(x) on one side of the equals sign, with a number on the other side.
To do this, get the two cosine terms on one side of the equation, and get the radical term on the other side.
The two cosine terms are like-terms, so they can be combined.
From there on, it's easy to solve for x, if you've memorized and understand the cosine values for basic angles.
Here's an example, using a similar equation.
\(\displaystyle 6 sin(x) = 4 sin(x) - \sqrt{2}\)
We want to isolate sin(x) on one side. So, we first get both sine terms to one side. We can do this by subtracting 4sin(x) from both sides.
\(\displaystyle 6 sin(x) - 4 sin(x) = 4 sin(x) - 4 sin(x) -\sqrt{2}\)
The terms 6sin(x) and -4sin(x) are like-terms. They can be combined.
\(\displaystyle 2 sin(x) = -\sqrt{2}\)
We don't want 2sin(x). We want to isolate sin(x). We can do this by dividing both sides by 2.
\(\displaystyle \frac{2}{2} \ sin(x) = \frac{-\sqrt{2}}{2}\)
\(\displaystyle sin(x) = -\frac{\sqrt{2}}{2}\)
Now, we solve for x by using our knowledge of the sine function.
Since we've memorized the value of sine for the basic angles, we know right away that the sine of 45 degrees is ?2/2.
Since we also understand that sine is negative in Quadrant III and Quadrant IV, we use 45 degrees as a reference angle in each of those quadrants.
180 + 45 = 225
360 - 45 = 315
So, there are two solutions for x (in this example) between 0 degrees and 360 degrees.
x = 225 degrees or x = 315 degrees
Please try the same strategy, with your exercise.
If I wrote anything that you do not understand, then please ask specific questions.
Cheers ~ Mark
In other words, you want to isolate the expression cos(x) on one side of the equals sign, with a number on the other side.
To do this, get the two cosine terms on one side of the equation, and get the radical term on the other side.
The two cosine terms are like-terms, so they can be combined.
From there on, it's easy to solve for x, if you've memorized and understand the cosine values for basic angles.
Here's an example, using a similar equation.
\(\displaystyle 6 sin(x) = 4 sin(x) - \sqrt{2}\)
We want to isolate sin(x) on one side. So, we first get both sine terms to one side. We can do this by subtracting 4sin(x) from both sides.
\(\displaystyle 6 sin(x) - 4 sin(x) = 4 sin(x) - 4 sin(x) -\sqrt{2}\)
The terms 6sin(x) and -4sin(x) are like-terms. They can be combined.
\(\displaystyle 2 sin(x) = -\sqrt{2}\)
We don't want 2sin(x). We want to isolate sin(x). We can do this by dividing both sides by 2.
\(\displaystyle \frac{2}{2} \ sin(x) = \frac{-\sqrt{2}}{2}\)
\(\displaystyle sin(x) = -\frac{\sqrt{2}}{2}\)
Now, we solve for x by using our knowledge of the sine function.
Since we've memorized the value of sine for the basic angles, we know right away that the sine of 45 degrees is ?2/2.
Since we also understand that sine is negative in Quadrant III and Quadrant IV, we use 45 degrees as a reference angle in each of those quadrants.
180 + 45 = 225
360 - 45 = 315
So, there are two solutions for x (in this example) between 0 degrees and 360 degrees.
x = 225 degrees or x = 315 degrees
Please try the same strategy, with your exercise.
If I wrote anything that you do not understand, then please ask specific questions.
Cheers ~ Mark