How many solutions do sin(t) = -1/7 have? I don't understand how to get the solutions
The answer depends upon the limits of t. If you mare talking about one cycle from t = 0 to t = 2 pi (or 360 degrees), there are two solutions. One is in the third quadrant and one is in the fourth.
The solutions are 351.8 degrees and 188.2 degrees. If you need it in radians, do the usual conversion.
If I remember the sin curve, it goes up and down forever, so the answer, is an infinite number of soltions. Now if you are interested in the solutions between t=0 and 2PI, it goes through that value twice.
Draw the sin curve on your calculator. Look to see where it is near -.14.... You should see two values. Get that curve memorized in your head, then you will think the approximate solutions. Using the small angle spproximation, sin=.14 at about .14 radian, or about 7 degrees. Since you want negative, it is in the third and fourth quadrant, about -7 deg, or 187 deg. You ought to be able to approximate these in your head. Memorize that curve, then use your calulator to get it exact. Check it.
To find the solutions to the equation sin(t) = -1/7, we can use the inverse sine function, also called arcsine or sin^(-1) (pronounced "sine inverse"). The inverse sine function allows us to find the angles that correspond to a given sine value.
Here's how you can find the solutions:
1. Start by writing the equation sin(t) = -1/7.
2. Take the inverse sine (arcsine) of both sides of the equation to isolate the variable t. The inverse sine of -1/7 can be written as sin^(-1)(-1/7) or arcsin(-1/7).
3. Using a calculator or a table of inverse trigonometric functions, find the value of arcsin(-1/7). This will give you one of the solutions.
4. Typically, inverse trigonometric functions return a principal value that falls within a specific range (-π/2 to π/2 for arcsine). However, since sin(t) can have multiple solutions for a given value, we need to consider all possible solutions.
5. To find the other solutions, we need to look at the symmetry and periodicity of the sine function. The sine function is periodic with a period of 2π, which means it repeats its values every 2π units. Additionally, its values are symmetric about the y-axis (i.e., sin(-x) = -sin(x)).
6. To find the other solutions, we can add or subtract multiples of 2π to the initial solution we found in step 3. This takes advantage of the periodicity of the sine function.
7. Continue adding or subtracting multiples of 2π to the initial solution to find all the solutions.
Keep in mind that the solutions will be in radians, as trigonometric functions typically use radians as their unit of measurement.
In summary, to find the solutions to sin(t) = -1/7, use the inverse sine function (arcsin) to find one solution and then account for the periodicity of the sine function by adding or subtracting multiples of 2π.