hi, i need help solving this trigonometric equation:
sinx+3cosx=0
i don't know what to do as i can't just diviide the two to create tanx which is what i have been doing in previous questions...
yes, you are on the right track
sinx = -3cosx
sinx/cosx = -3
tanx = -3
so x must be in the II and IV quadrant by the CAST rule
take arctan(+3) to get 71.56º
then x=180º-71.56 = 108.44º
or
x = 360-71.56 = 288.44º
check them with a calculuator, they both work
To solve the trigonometric equation sin(x) + 3cos(x) = 0, you're correct that you can't simply divide the two terms like in previous questions. However, you can rearrange the equation to obtain an expression involving tangent (tan). Here's how you can do it:
1. Start with the equation: sin(x) + 3cos(x) = 0.
2. Divide both sides of the equation by cos(x) (assuming cos(x) is not equal to zero) to eliminate the cosine term: (sin(x)/cos(x)) + 3 = 0.
3. Simplify the expression sin(x)/cos(x) to obtain tan(x): tan(x) + 3 = 0.
4. Subtract 3 from both sides of the equation: tan(x) = -3.
Now that you have tan(x) = -3, you can use the inverse tangent function (arctan) to find the values of x that satisfy this equation. Keep in mind that the tangent function has a periodicity of π radians (180 degrees), so we need to consider all possible angles within one period.
1. Take the arctan of both sides of the equation: arctan(tan(x)) = arctan(-3).
2. Applying the arctan to both sides simplifies the equation to: x = arctan(-3).
3. Use a calculator or a table of trigonometric values to find the principal value of arctan(-3), which is approximately -71.56 degrees.
Now, since the tangent function is negative in the second (II) and fourth (IV) quadrants, we have two potential solutions for x. To find them:
4. In the second quadrant, add 180 degrees to the principal value: x = -71.56 degrees + 180 degrees = 108.44 degrees.
5. In the fourth quadrant, subtract the principal value from 360 degrees: x = 360 degrees - 71.56 degrees = 288.44 degrees.
Both x = 108.44 degrees and x = 288.44 degrees should satisfy the original equation sin(x) + 3cos(x) = 0. You can verify this using a calculator by substituting these values back into the equation.