# What is tanx if 2cosx sinx 1

## Equations with sine, cosine and tangent

### What is a trigonometric equation?

A trigonometric equation is an equation in which at least one trigonometric functionSine, cosine or tangent occurs.

To solve such equations, you need a calculator. Make sure this is on DEG For degree, so Angle measureis set.

### \$ \ sin (x) = c \$

A trigonometric equation is given for example by \$ \ sin (x) = 0.5 \$. So all values ​​for \$ x \$ are searched for, for which \$ f (x) = \ sin (x) = 0.5 \$. Look at the graph of the function \$ f (x) = \ sin (x) \$. • To get a solution to the above equation, use the inverse function of \$ \ sin (x) \$ on the calculator, den Arcsine \$ \ sin ^ {- 1} \$ or \$ \ arcsin \$.
• A solution to the equation is then \$ x_1 = sin ^ {- 1} (0.5) = 30 ^ \ circ \$.
• The calculator always outputs values ​​between \$ -90 ^ \ circ \$ and \$ 90 ^ \ circ \$ for equations of the form \$ \ sin (x) = c \$, with \$ c \ in [-1; 1] \$.
• As you can see from the function graph, there is another solution. You get this by subtracting the solution given by the calculator from \$ 180 ^ \ circ \$, i.e. \$ 30 ^ \ circ \$: \$ x_2 = 180 ^ \ circ-30 ^ \ circ = 150 ^ \ circ \$.
• The thus obtained Solution pair \$ x_1 = 30 ^ \ circ \$ and \$ x_2 = 150 ^ \ circ \$ are called Basic solution designated.
• Due to the \$ 360 ^ \ circ \$ -periodicity the Sine function then all solutions of the equation are given by:

\$ \ quad ~~~ x_1 ^ {(k)} = 30 ^ \ circ + k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$ and

\$ \ quad ~~~ x_2 ^ {(k)} = 150 ^ \ circ + k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$.

Similarly, if there is a negative number on one side of the equation, you get all the solutions: \$ \ sin (x) = - 0.5 \$.

• Then \$ x_1 = \ sin ^ {- 1} (- 0.5) = - 30 ^ \ circ \$.
• The other Basic solution then \$ x_2 = -180 ^ \ circ + 30 ^ \ circ = -150 ^ \ circ \$.
• Here, too, you get the totality of the solution with the help of the periodicity.

\$ \ quad ~~~ x_1 ^ {(k)} = -30 ^ \ circ-k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$ and

\$ \ quad ~~~ x_2 ^ {(k)} = -150 ^ \ circ-k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$.

### \$ \ cos (x) = c \$

• The calculator always outputs values ​​between \$ 0 ^ \ circ \$ and \$ 180 ^ \ circ \$ for equations of the form \$ \ cos (x) = c \$, with \$ c \ in [-1; 1] \$.
• The other Basic solution you get by swapping the sign.
• Here, too, you can use the periodicity the cosine function.

example: \$ \ cos (x) = \ frac1 {\ sqrt2} \$

Then

\$ x_1 = \ cos ^ {- 1} \ left (\ frac1 {\ sqrt2} \ right) = 45 ^ \ circ \$.

Now \$ x_2 = -45 ^ \ circ \$ and

\$ \ quad ~~~ x_1 ^ {(k)} = 45 ^ \ circ + k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$ and

\$ \ quad ~~~ x_2 ^ {(k)} = - 45 ^ \ circ + k \ cdot 360 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$.

### \$ \ tan (x) = c \$

• The Tangent function is \$ 180 ^ \ circ \$ -periodically.
• The calculator outputs an angle between \$ -90 ^ \ circ \$ and \$ 90 ^ \ circ \$. (Note that the tangent is not defined for either \$ 90 ^ \ circ \$ or \$ -90 ^ \ circ \$.)

example: \$ \ tan (x) = 1 \$

• The calculator solution is \$ x = \ tan ^ {- 1} (1) = 45 ^ \ circ \$.
• The set of solutions is then given by

\$ \ quad ~~~ x ^ {(k)} = 45 ^ \ circ + k \ cdot 180 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$.

### Trigonometric equations with two trigonometric functions and the same argument

How can you trigonometric equation solve in which two different trigonometric functions occur with the same argument?

\$ (\ cos (x)) ^ 3-2 \ cos (x) \ cdot \ sin ^ 2 (x) = 0 \$

• First you factor out \$ \ cos (x) \$.

\$ \ quad ~~~ \ cos (x) \ left (\ cos ^ 2 (x) -2 \ sin ^ 2 (x) \ right) = 0 \$

• A product becomes \$ 0 \$ when one of the factors becomes \$ 0 \$. So either \$ \ cos (x) = 0 \$ or \$ \ cos ^ 2 (x) -2 \ sin ^ 2 (x) = 0 \$.
• The zeros of \$ \ cos (x) \$ are \$ x = (2k + 1) \ cdot 90 ^ \ circ \$, \$ k \ in \ mathbb {Z} \$, i.e. the odd multiples of \$ 90 ^ \ circ \$.
• Now the second factor remains. Because of \$ \ sin ^ 2 (x) + \ cos ^ 2 (x) = 1 \$, this is the trigonometric pythagoras, \$ \ cos ^ 2 (x) = 1- \ sin ^ 2 (x) \$ and thus

\$ \ quad ~~~ 1- \ sin ^ 2 (x) -2 \ sin ^ 2 (x) = 1-3 \ sin ^ 2 (x) = 0 \$.

• You can transform this equation as follows.

\$ \ quad ~~~ \ begin {array} {rclll} 1-3 \ sin ^ 2 (x) & = & 0 & | & + 3 \ sin ^ 2 (x) \ 1 & = & 3 \ sin ^ 2 (x) & | &: 3 \ \ frac13 & = & \ sin ^ 2 (x) & | & \ sqrt {~~~} \ \ pm \ frac1 {\ sqrt3} & = & \ sin (x) & | & \ sin ^ {- 1} (~~~) \ \ pm35,3 ^ \ circ & \ approx & x \ end {array} \$

• For each of the two solutions, as above, you can first choose the missing one Basic solution determine and then the Total solution.

### Trigonometric equations with two trigonometric functions and different arguments

Such equation is given for example by

\$ \ cos (x) - \ sin \ left (\ frac x2 \ right) = 0 \$.

Not just two different ones dive here Trigonometric functions on, but also various arguments.

\$ \ quad ~~~ \ cos (x) = \ cos \ left (2 \ cdot \ frac x2 \ right) \$

\$ \ quad ~~~ \ cos \ left (2 \ cdot \ frac x2 \ right) = 1-2 \ sin ^ 2 \ left (\ frac x2 \ right) \$.

• With that, the above equation can be written as follows:

\$ \ quad ~~~ 1-2 \ sin ^ 2 \ left (\ frac x2 \ right) - \ sin \ left (\ frac x2 \ right) = 0 \$

• This is a quadratic function in \$ \ sin (x) \$. If you

\$ \ quad ~~~ z = \ sin \ left (\ frac x2 \ right) \$

Substituting \$ \ quad ~~~ \$, you get the quadratic equation \$ 1-2z ^ 2-z = 0 \$.

• You can do this with the p-q formula to solve. For this you put the equation around \$ -2z ^ 2-z + 1 = 0 \$ and divide by \$ -2 \$.

\$ \ quad ~~~ \ begin {array} {rclll} -2z ^ 2-z + 1 & = & 0 & | &: (- 2) \ z ^ 2 + \ frac12z- \ frac12 & = & 0 \ z_ {1, 2} & = & - \ frac14 \ pm \ sqrt {\ frac1 {16} + \ frac12} \ z_1 & = & - \ frac14 + \ frac34 = \ frac12 \ z_2 & = & - \ frac14- \ frac34 = -1 \ end {array} \$

• Finally, you resubstitute. So you need to solve the following equations:

\$ \ quad ~~~~ \ sin \ left (\ frac x2 \ right) = \ frac12 \$ and

\$ \ quad ~~~~ \ sin \ left (\ frac x2 \ right) = - 1 \$.

• Proceed as in the above examples for \$ \ sin (x) = c \$.