How would I solve this trigonometry function

Trigonometric equations

What are trigonometric equations?

In trigonometric equations the variable $ x $ occurs as an argument of a trigonometric function. These are for example Equations with sine, cosine and tangent.

Here is an example of a trigonometric equation:

$ \ sin (2x) -2 = -2.5 $.

When dealing with trigonometric equations, note whether the variable im Angle measure or in Radians is present. Depending on that, you have to set your calculator: $ D $ for DEG (Angle) or $ R $ for WHEEL (Radians).

The angle measure is used in the following.

Solving equations with sine or cosine

When solving equations with sine or cosine, you do the following:

  1. You isolate the term in which sine or cosine occurs.
  2. Then you turn the appropriate one Inverse function at.
  3. You determine the amount of solution.

For the second step you use the calculator. The inverse function of sine is that Arcsine, that of cosine of Arc cosine. In most cases you can find these above the corresponding keys on your calculator. The labels are $ \ sin ^ {- 1} $ or $ \ cos ^ {- 1} $.

Sine

Here is an example: You should solve the above equation $ \ sin (2x) -2 = -2.5 $. Here $ x \ in I = \ left [-90 ^ \ circ; 90 ^ \ circ \ right] $ should be.

$ \ begin {array} {rclll} \ sin (2x) -2 & = & - 2.5 & | & + 2 \ \ sin (2x) & = & - 0.5 & | & \ sin ^ {- 1} ( ~~~) \ 2x & = & {- 30} ^ \ circ & | &: 2 \ x & = & {- 15} ^ \ circ \ end {array} $

You have found a solution with $ x_ {1} = {- 15} ^ \ circ $.

Here you can see the green function graph of the function $ y = \ sin (2x) -2 $. The dashed line runs parallel to the $ x $ axis through $ y = -2.5 $. The interfaces between the function graph and this line are the solutions sought.

Because of the symmetry of the sine function, you get another solution. This is $ x_ {2} = {- 90} ^ \ circ + 15 ^ \ circ = {- 75} ^ \ circ $. These solutions are the only ones in the interval given above. You can see these solutions in the picture.

In addition, you can see that there are other places outside the interval where the function graph intersects the dashed line.

Since $ \ sin (2x) $ has a $ 180 ^ \ circ $ periodicity, there are infinitely many solutions. Another solution is, for example, $ x_ {3} = -15 ^ \ circ + 180 ^ \ circ = 165 ^ \ circ $.

To give all solutions, we define a variable $ k \ in \ mathbb {Z} $. The following then applies to the solutions:

  • $ x_ {1} ^ {k} = {- 15} ^ \ circ + k \ cdot 180 ^ \ circ $
  • $ x_ {2} ^ {k} = {- 75} ^ \ circ + k \ cdot 180 ^ \ circ $

Note: The $ k $ is in this case no exponent just a numbering.

Cosine

Find all solutions of the equation $ \ cos (x) = 0.5 $. Also in this example you consider the interval $ I = \ left [-90 ^ \ circ; 90 ^ \ circ \ right] $.

You can see the cosine function and a dashed parallel to the $ x $ axis through $ y = 0.5 $. The solutions are the interfaces of the cosine function with these parallels.

With the calculator you get $ x_ {1} = \ cos ^ {- 1} (0.5) = 60 ^ \ circ $. You get another solution with the symmetry of the cosine function. $ X_ {2} = {- 60} ^ \ circ $ applies. These two solutions lie in the interval considered.

As with the equation above, you can find more solutions this time too. The cosine function is $ 360 ^ \ circ $ -periodic. You will get the following solutions:

  • $ x_ {1} ^ {k} = {60} ^ \ circ + k \ cdot 360 ^ \ circ $
  • $ x_ {2} ^ {k} = {- 60} ^ \ circ + k \ cdot 360 ^ \ circ $

Again the variable is $ k \ in \ mathbb {Z} $. The superscript $ k $ is also used here for numbering.

Solving equations with tangent

Find all solutions of the equation $ 2 \ tan (3x) + 1 = 3 $. First isolate $ \ tan (3x) $:

$ \ begin {array} {rclll} 2 \ tan (3x) +1 & = & 3 & | & -1 \ 2 \ tan (3x) & = & 2 & | &: 2 \ \ tan (3x) & = & 1 \ end {array} $

Now you get a solution from the calculator with $ 3x = \ tan ^ {- 1} (1) = 45 ^ \ circ $. The tangent function is $ 180 ^ \ circ $ -periodic. Overall, this leads to the solution set $ 3x ^ {k} = 45 ^ \ circ + k \ cdot 180 ^ \ circ $. Division by $ 3 $ finally leads to $ x ^ {k} = 15 ^ \ circ + k \ cdot 60 ^ \ circ $ with $ k \ in \ mathbb {Z} $.

Note: Just like above, $ k $ is here no exponent.

The addition therems

For example, if a term of the form $ \ sin (x + 45 ^ \ circ) $ occurs in a trigonometric equation, you can use a Addition theorem use to transform this term.