How Do You Find Multiple Solutions for Trigonometric Equations?

[bsmithysmith]

I have a quiz on Friday and I'm not understanding one part of the section;

find all the solutions for \(\displaystyle sin(Ø)=0.8\)

and I'm not understanding how to find the other solution. This may be an easier one, but there's also:\(\displaystyle Sin(Ø)=-.58\)
\(\displaystyle Cos(Ø)=-.55\)
\(\displaystyle Cos(Ø)=.71\)Need help finding two solutions for each!

 

[MarkFL]

For $\sin(\theta)=0.8$, draw a unit circle, and then draw the line $y=0.8$. Now, from the center of the unit circle, draw two rays, one to each point of intersection between the circle and the line. The angle subtended by each ray and the positive $x$-axis in the counter-clockwise direction are your two solutions, given that we are looking for solutions in:

\(\displaystyle 0\le\theta<2\pi\)

For the first quadrant angle, use you calculator to find an approximation for:

\(\displaystyle \theta=\sin^{-1}(0.8)\)

Now, can you see how to use this value to determine the second quadrant solution? It has to do with the identity:

\(\displaystyle \sin(\pi-\theta)=\sin(\theta)\)

From your drawing, can you "see" how this identity has to be true?

 

[bsmithysmith]

For $\sin(\theta)=0.8$, draw a unit circle, and then draw the line $y=0.8$. Now, from the center of the unit circle, draw two rays, one to each point of intersection between the circle and the line. The angle subtended by each ray and the positive $x$-axis in the counter-clockwise direction are your two solutions, given that we are looking for solutions in:

\(\displaystyle 0\le\theta<2\pi\)

For the first quadrant angle, use you calculator to find an approximation for:

\(\displaystyle \theta=\sin^{-1}(0.8)\)

Now, can you see how to use this value to determine the second quadrant solution? It has to do with the identity:

\(\displaystyle \sin(\pi-\theta)=\sin(\theta)\)

From your drawing, can you "see" how this identity has to be true?

SWEET! Helped a bunch! Thank You!

 

[MarkFL]

For the remaining problems use a similar technique, except where you have the cosine function, use a vertical line instead of a horizontal line for the intersections to help you visualize where the solution(s) are. :D

 

[CellCoree]

As a scientist, my advice would be to use the unit circle and trigonometric identities to find the solutions for sin(Ø)=0.8. The unit circle is a useful tool for understanding the relationship between sine, cosine, and other trigonometric functions. You can also use the inverse sine function (arcsine) to find the solutions.

For sin(Ø)=0.8, the solutions are Ø=53.13° and Ø=126.87°. This can be found by using the inverse sine function on a calculator or by using the unit circle and remembering that the sine function is positive in the first and second quadrants.

For the second set of equations, you can use the Pythagorean identity (sin^2Ø+cos^2Ø=1) to find the missing values. For example, for sin(Ø)=-0.58, you can square both sides and then subtract from 1 to find cos^2Ø=0.65. Taking the square root of both sides, you get cos(Ø)=±0.81. Similarly, for cos(Ø)=-0.55, you can use the same method to find sin(Ø)=±0.83. For cos(Ø)=0.71, you can use the inverse cosine function to find the solutions Ø=44.42° and Ø=315.58°.

In general, it is helpful to use the unit circle and trigonometric identities to solve for trigonometric equations. Practice using these tools and you will become more comfortable finding solutions for different equations. Good luck on your quiz!