Question

Как найти cosx, sinx, если известен tgx, и ctgx?

Answer 1

Тангенс это sinx:cosx
А ctg=cosx:sinx

Answer 2

Finding cos(x) and sin(x) given tg(x) and ctgx

To find the values of cos(x) and sin(x) when tg(x) and ctgx are known, we can use trigonometric identities and relationships.

First, let's recall some basic trigonometric identities: - sin(x) is the ratio of the length of the side opposite angle x to the length of the hypotenuse in a right triangle. - cos(x) is the ratio of the length of the side adjacent to angle x to the length of the hypotenuse in a right triangle. - tg(x) is the ratio of sin(x) to cos(x), which can be expressed as tg(x) = sin(x) / cos(x). - ctgx is the reciprocal of tgx, which can be expressed as ctgx = cos(x) / sin(x).

Using these identities, we can solve for cos(x) and sin(x) in terms of tg(x) and ctgx.

Solving for cos(x)

From the identity tg(x) = sin(x) / cos(x), we can rearrange the equation to solve for cos(x):

cos(x) = sin(x) / tg(x) [[8]](https://mipt.ru/education/chair/mathematics/study/uchebniki/%D0%94%D1%8B%D0%BC%D0%B0%D1%80%D1%81%D0%BA%D0%B8%D0%B9_%D0%9C%D0%90_1(4).pdf).

Solving for sin(x)

From the identity ctgx = cos(x) / sin(x), we can rearrange the equation to solve for sin(x):

sin(x) = cos(x) / ctgx [[8]](https://mipt.ru/education/chair/mathematics/study/uchebniki/%D0%94%D1%8B%D0%BC%D0%B0%D1%80%D1%81%D0%BA%D0%B8%D0%B9_%D0%9C%D0%90_1(4).pdf).

Now, let's use these equations to find the values of cos(x) and sin(x) given tg(x) and ctgx.

Example:

Let's say tg(x) = 2 and ctgx = 3.

Using the equations above, we can find the values of cos(x) and sin(x):

cos(x) = sin(x) / tg(x) = sin(x) / 2

sin(x) = cos(x) / ctgx = cos(x) / 3

Since we have two equations with two unknowns (cos(x) and sin(x)), we can solve them simultaneously.

Let's substitute sin(x) / 2 for cos(x) in the second equation:

sin(x) = (sin(x) / 2) / 3

Now, we can solve for sin(x):

sin(x) = sin(x) / 6

To solve this equation, we can multiply both sides by 6:

6 * sin(x) = sin(x)

Subtract sin(x) from both sides:

6 * sin(x) - sin(x) = 0

Simplifying the equation:

5 * sin(x) = 0

Divide both sides by 5:

sin(x) = 0

Therefore, sin(x) = 0.

Now, let's substitute sin(x) = 0 into the equation cos(x) = sin(x) / 2:

cos(x) = 0 / 2

cos(x) = 0

Therefore, cos(x) = 0.

So, when tg(x) = 2 and ctgx = 3, the values of cos(x) and sin(x) are both 0.

Please note that this is just an example, and the values of tg(x) and ctgx can vary in different scenarios.

I hope this helps! Let me know if you have any further questions.