# Solving Trigonometric Problems

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The first solution we get comes from the inverse sine.

Inverse sine of a half is going to give us the angle in between -5 over 2 and pi over 2 that's satisfies the equation in this case it pi over 6 this solution.

You may wish to go back and have a look at Trigonometric Functions of Any Angle, where we see the background to the following solutions. Where the graph cuts the x-axis, that's where you'll find your solutions (the x-values that "work").

But you could see that within the interval from 0 to 2pi within the first period of 2 of of sine sine theta there are two solutions here's the second one. We [IB] probably see by symmetry that this angle here is theta so this angle will have to be pi minus theta, the supplement so that's the second solution.

Always remember this identity the sine of pi minus x equals the sine of x with the sine function supplementary angles have the same sine value so an important property of the sine function so if pi over 6 works, 5pi over 6 is also a solution right?Trigonometry is the study of relationships that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures.Applications of trigonometry are also found in engineering, astronomy, Physics and architectural design.It will have infinitely many points and there are going to be two points per period so expect infinitely many answers and expect to have two per period going into the problem.Now usually I actually find the solutions on the unit circle, so I've drawn a unit circle and I've also drawn the line y equals one half because remember if I draw an angle the point on the unit circle where the angle crosses that point p its y coordinate is going to be the sine of this angle so in this case the y coordinate it's going to have to be one half so the question is what is this angle theta?Now let us start with the basic formulas of trigonometry and see the basic relationships on which the whole concept is based on.In a right-angled triangle, we have Hypotenuse, Base and Perpendicular.Two ways to visualize the solutions are (1) the graph in the coordinate plane and (2) the unit circle.The unit circle is the more useful of the two in obtaining an answer. Let's start with a really simple example, sine of theta equals a half.When solving trigonometric equations, we find all the angles that make the equation true.If there is no interval given, use periodicity to show the infinite number of solutions.

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