MHB 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 $$ sin(Ø)=0.8$$

and I'm not understanding how to find the other solution. This may be an easier one, but there's also:$$Sin(Ø)=-.58$$
$$Cos(Ø)=-.55$$
$$Cos(Ø)=.71$$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:

$$0\le\theta<2\pi$$

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

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

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

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

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:

$$0\le\theta<2\pi$$

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

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

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

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

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