Given that sin x = - �ã2/2 and that cos x is negative, find the other functions of x and the value of x.

can you explain how this problem can be solved please.

Do you mean -(sqrt(2))/2 ?

yes

okay i know that this is in the 3rd quadrant with reference angle 45
so i understand the first part but what i don't get is how i can find the value can you please explain.

sides of triangle are sqrt 2, sqrt 2 , 2 so

cos x also = -sqrt 2 / 2
tan = sin / cos
sec = 1/cos
csc = 1/sin
ctn = 1/tan

sin = -sqrt 2 / 2

cos = -sqrt 2 / 2
tan = 1
cot = 1
sec = -sqrt 2
csc = -sqrt 2

is that right ?
okay now how can i find the value?

yes. Sqrt 2 is approximately 1.41

x = 180 + 45 in degrees

= (1 1/4) pi in radians

thanks for your time but they are also asking for the value of x which i think they mean the value of x in degrees ??

thank you.

x = 225 degrees

To solve this problem, we can use the given information about the value of sin x and the negativity of cos x to determine the values of the other trigonometric functions and the actual value of x.

First, we are given that sin x = -√2/2. From this, we can determine the angle whose sine is -√2/2 by looking at the unit circle or using special angles. We know that sin 45° = √2/2. Since sin is negative, we can conclude that the angle x is either in the third or fourth quadrant.

Next, we are told that cos x is negative. From this, we can further narrow down the possibilities for the angle x. In the unit circle, cos is negative in the second and third quadrants.

Combining the information that sin x is negative and cos x is negative, we find that x is in the third quadrant. In the third quadrant, sin is negative, and cos is also negative.

Now, let's find the values of the other trigonometric functions. In the third quadrant, sin is negative, so sin x = -√2/2 is confirmed. Cos is negative, so cos x = -√2/2.

To find the other trigonometric functions, we can use the values of sin x and cos x to determine the values of tan x, cot x, sec x, and csc x.

Using the definition of tangent (tan x = sin x / cos x), we can calculate tan x = (-√2/2) / (-√2/2) = 1.

Using the definition of cotangent (cot x = cos x / sin x), we can calculate cot x = (-√2/2) / (-√2/2) = 1.

Using the definition of secant (sec x = 1 / cos x), we can calculate sec x = 1 / (-√2/2) = -√2.

Using the definition of cosecant (csc x = 1 / sin x), we can calculate csc x = 1 / (-√2/2) = -√2.

Therefore, the values of the other trigonometric functions are tan x = 1, cot x = 1, sec x = -√2, and csc x = -√2.

To find the actual value of x, we need to determine the angle in the third quadrant that has the same sine value as -√2/2. Since sin 45° = √2/2, we know that the reference angle in the third quadrant is 180° - 45° = 135°.

Therefore, x = 135°.

In summary, the values of the other trigonometric functions of x are tan x = 1, cot x = 1, sec x = -√2, and csc x = -√2. The value of x is 135°.