Use The Given Point In The Terminal Side Of Angle Theta To Find The Value Of The Trigonometric

1) Using a triangle..Step 1:Using a compass or protractor, draw the angle you need to measure the trigonometric function of.Complete the triangle by dropping a perpendicular from any point of one ray on to the other.Measure the side lengths using a ruler.Now,[math]\frac{P}{H} = sin\theta[/math][math]\frac{B}{H} = cos\theta[/math][math]\frac{P}{B} = tan\theta[/math][math]\frac{H}{P} = cosec\theta[/math][math]\frac{H}{B} = sec\theta[/math][math]\frac{B}{P} = cot\theta[/math]Similarly, to compute inverse trigonometric functions, first draw the sides, complete the triangle, and then measure the concerning angle to get the result.2) By Taylor Series[math]cos(x) = 1 - \frac{x^2}{2!}   + \frac{x^4}{4!}  - \frac{x^6}{6!} + ...[/math][math]sin(x) = x - \frac{x^3}{3!}   + \frac{x^5}{5!}  - \frac{x^7}{7!} + ...[/math][math]tan(x) = x + x^3 + \frac{2x^5}{15} + ...[/math][math]sin^{-1}(x) = x + \frac{x^3}{6} + \frac{3x^5}{40} + ...[/math]Substitute the argument of the function (x) into the series, and calculate its value. The more terms you use, the more accurate your answer.If you aren't familiar with the concept of Taylor Series, then to put it in a nutshell, any function can be written in the form of a polynomial expansion. That polynomial is called its Taylor Series.If you want to know how to find the series of any function, I suggest you read this answer or watch this video: https: //youtu . be/zdJl27kgeSQ.3) By InterpolationThere are several methods to interpolate functions. Interpolation basically means finding the value of a function at a point, when you already know the value of that function at other points.So suppose you know the value of [math]sin(0), sin(\frac{\pi}{6}), sin(\frac{\pi}{4}), sin(\frac{\pi}{3}), sin(\frac{\pi}{2})[/math]. That means, that using these values, you can figure out the value of say sin(0.532), by assuming a pattern from these known values. The more values you know, the more accurate your answer, because the pattern you're assuming becomes closer to the actual function.There are many methods of interpolation such as:Lagrange: http://en.wikipedia.org/wiki/Lag...Finite differences: http://en.wikipedia.org/wiki/Fin...Runge-Kutta: http://en.wikipedia.org/wiki/Run...I'll take the example of Lagrange interpolation:where [math]f(x_0), f(x_1), ..... f(x_k)[/math] are known.Substitute the known values, and find the value of L(x), which is what was required.Thanks for the A2A!

Let’s first consider the unit circle, i.e., a circle with a radius of length r = 1 unit and with its center at the origin (0,0) of a rectangular coordinate system (the xy-plane); therefore, the unit circle intersects the x-axis of the rectangular coordinate system at the points (1,0) and (–1,0) and intersects the y-axis at the points (0,1) and (0, –1).Let the radius of the unit circle be a line segment OP whose length is r = 1 unit and that extends from the origin O (0,0) and intersects the unit circle at point P (x,y) on the circle. Let angle θ be the angle that radius OP of the unit circle makes with the positive x-axis. Also, let angle θ be any angle in standard position with its vertex at the origin O (0,0), with its initial side along the positive x-axis, and with radius OP as its terminal side, which, by the way, can lie in any one of the four (4) quadrants, and angle θ can be either positive (counterclockwise rotation) or negative (clockwise rotation). The six basic trigonometric functions such as the sine function of angle θ (sin θ) and the cosine function of angle θ (cos θ) express the interrelationships that exist between angle θ, radius length r, and the coordinates (x, y) of point P. The cosine function of angle θ is defined by a ratio which is expressed as cos θ = x/r = (abscissa of point P)/(length of radius OP); however, since we’re dealing with a unit circle where r = 1, we have: cos θ = x/r           = x/1 1.) cos θ  = x,  and …The sine function of angle θ is also defined by a ratio which is expressed as sin θ = y/r = (ordinate of point P)/(length of radius OP); however, since we’re dealing with a unit circle where r = 1, we have: sin θ = y/r          = y/12.) sin θ = yThus, the above discussion explains and illustrates how the cosine (cos) and sine (sin) functions translate to x and y on the unit circle.

If you want to represent an angle using a trigonometric circle you need to "trim" it down to value that's between [math]\left[0,2 \pi\right)[/math].Notice the last bracket, it means that you aren't including the [math],2 \pi[/math], since that's the same angle as 0. Any angle can be represented in a form [math]2k\pi+\theta[/math], where [math]\theta[/math] is the angle in the trigonometric circle, and k is an integer. This angle is measured starting from the x axis going counter clockwise. The quadrants are defined to be:I [math]\left[0, \frac{\pi}{2}\right)[/math]II [math]\left[\frac{\pi}{2},\pi \right)[/math]III [math]\left[\pi,\frac{3\pi}{2} \right)[/math]IV [math]\left[\frac{3\pi}{2},2\pi \right)[/math]Adding  [math]2k\pi[/math] to an angle doesn't change it's position so that can be "trimmed down" by removing that until the resulting angle is in the interval [math]\left[0,2 \pi\right)[/math].This is practically a module you get by dividing the angle with [math]2 \pi[/math].In your case the results will be:[math]\frac{\pi}{2}<\frac{5\pi}{6}<\pi[/math], this means that the angle is in the second quadrant. As for [math]\frac{107\pi}{6}[/math] first we need to do the trimming.The easiest way to do this is to, firstly, divide [math]\frac{107\pi}{6}[/math] with [math]2 \pi[/math]. We get 107/12 which is 8 and mod is 11. This means that we can write [math]\frac{107\pi}{6}=8\cdot2\pi+11\frac{\pi}{6}[/math].Since the first part is a multiple of [math]2\pi[/math] it doesn't affect the place in the circle, so we just need to look at that second term.We have that:[math]\frac{3\pi}{2}<\frac{11\pi}{6}<2\pi[/math], meaning that it's in the 4th quadrant.If the angle is negative then counting goes clock wise, meaning that if the absolute value is in the 1st quadrant then the angle will be in the 4th, 2nd means the 3rd, 3rd is the 2nd and the 4th is the 1st.