subject: Trigonometric Solutions [print this page] Trigonometric means measurement of trianglesTrigonometric means measurement of triangles. Trigonometric are mainly used in triangle for finding the angles and side lengths. In trigonometric, we have the different functions. They are general trigonometric functions, inverse trigonometric functions, hyperbolic trigonometric functions, and inverse hyperbolic trigonometric functions. Trigonometric solutions are derived by using the above four functions. Trigonometric functions are also used for geometry for finding the angles.
Trigonometric Equations
Trigonometric identities are true for all replacement values for the variables for which both sides of the equation are defined. Conditional trigonometric equations are true for only some replacement values. Solutions in a specific interval, such as 0 x 2, are usually called primary solutions. A general solution is a formula that names all possible solutions.
The process of solving general trigonometric equations is not a clear-cut one. No rules exist that will always lead to a solution. The procedure usually involves the use of identities, algebraic manipulation, and trial and error. The following guidelines can help lead to a solution.
If the equation contains more than one trigonometric function, use identities and algebraic manipulation (such as factoring) to rewrite the equation in terms of only one trigonometric function. Look for expressions that are in quadratic form and solve by factoring. Not all equations have solutions, but those that do usually can be solved using appropriate identities and algebraic manipulation.
Example Problems for Trigonometric Function:
Example 1:
Prove that the trigonometric function (1 + sin) / (1 - sin) = (sec + tan)2
Solution:
First, take the R.H.S value,
(sec + tan) = [(1 / cos) + (sin / cos)]2
= (1 + sin)2 / cos2
= (1 + sin) (1 + sin) / (1 + sin) (1 - sin)
= 1 + sin / 1 - sin
Therefore, R.H.S = L.H.S, Hence proved.
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Example 2:
Find the trigonometric solution for the given function y = cot x
Solution:
Rearrange the given function as, we get
y = cot x = (cos x / sin x)
Differentiate with respect to x, using (u / v) formula, we get
dy / dx = [sinx (- sinx) - cosx cosx / sin2x]
= (- sin2x + cos2x) / sin2x
= - 1 / sin2x
= - cosec2x
Answer:
The final solution is - cosec2x
Example 3:
Solve the trigonometric function y = sin3x / cos 2x
Solution:
Differentiate the given equation with respect to x , use (u / v) formula, we get
dy /dx = cos 2x (3 cos 3x) - sin 3x (- 2 sin 2x) / (cos2 2x)
= (3 cos 2x cos 3x + 2 sin 3x sin 2x) / (cos2 2x)
Answer:
The final answer is (3 cos2x cos3x + 2 sin3x sin 2x) / (cos22x)
Trigonometric Solutions-practice Problems
1) Solve the trigonometric function and find its solution y = sin 4x - cos 5x
Answer:
The final solution is 4 cos 4x + 5 sin 5x
2) Solve the trigonometric function and find its solution y = cos x sin 2x
Answer:
The final answer is 2 cos x cos 2x - sin x sin 2x
3) Solve the trigonometric function sin2 x (3x - 2)
