Respuesta :
Ok, you are given a point and you need to find the exact values for [tex]cos \theta csc \theta & tan \theta[/tex]
First thing first. We need to see if we are working with a unit circle and find the radius.
How to tell if we are working with a unit circle?
We know [tex]x^2 + y^2 = r^2[/tex] is a circle.
We know that to find the radius we can use the following formula:
[tex]r^2 = \sqrt{x^2 + y^2}[/tex]
If [tex]r^2 = \sqrt{x^2 + y^2} = 1[/tex] we are working with a unit circle.
Lets see if it = 1.
[tex]r^2 = \sqrt{4^2 + -7^2}[/tex]
[tex]r^2 = \sqrt{16 + 49}[/tex]
[tex]r^2 = \sqrt{65}[/tex]
Square both sides now
[tex]\sqrt{r^2} = \sqrt{\sqrt{65}}[/tex]
[tex]r = \pm 65^{\frac{1}{4}}}[/tex]
Since we squared, we have a + and a - but we disregard the - because we do not have - radii
[tex]r = 65^{\frac{1}{4}}}[/tex]
We can also say
[tex]r = \sqrt[4]{65}[/tex]
Ok, since r does not equal 1, we are not working with a unit circle but we have found r, which is our radius.
Now that we know the value of r, which is [tex]r = 65^{\frac{1}{4}}}[/tex], we need to look at the identities of cos, csc and tan.
The identities:
[tex]cos \theta = \frac{x}{r}[/tex]
[tex]csc \theta = \frac{1}{y}[/tex]
[tex]tan \theta = \frac{y}{x}[/tex]
Now that we know their identities and know the radius of our circle, we can find the exact values of cos, csc and tan.
[tex]cos \theta = \frac{x}{r} = \frac{4}{65^{\frac{1}{4}}}[/tex]
[tex]csc \theta = \frac{1}{y} = \frac{1}{-7}[/tex]
[tex]tan \theta = \frac{y}{x} = \frac{-7}{4}[/tex]
The exact values for cos, csc and tan given the point (4,-7) are:
[tex]cos \theta = \frac{4}{65^{\frac{1}{4}}}[/tex]
[tex]csc \theta = \frac{1}{-7}[/tex]
[tex]tan \theta = \frac{-7}{4}[/tex]
First thing first. We need to see if we are working with a unit circle and find the radius.
How to tell if we are working with a unit circle?
We know [tex]x^2 + y^2 = r^2[/tex] is a circle.
We know that to find the radius we can use the following formula:
[tex]r^2 = \sqrt{x^2 + y^2}[/tex]
If [tex]r^2 = \sqrt{x^2 + y^2} = 1[/tex] we are working with a unit circle.
Lets see if it = 1.
[tex]r^2 = \sqrt{4^2 + -7^2}[/tex]
[tex]r^2 = \sqrt{16 + 49}[/tex]
[tex]r^2 = \sqrt{65}[/tex]
Square both sides now
[tex]\sqrt{r^2} = \sqrt{\sqrt{65}}[/tex]
[tex]r = \pm 65^{\frac{1}{4}}}[/tex]
Since we squared, we have a + and a - but we disregard the - because we do not have - radii
[tex]r = 65^{\frac{1}{4}}}[/tex]
We can also say
[tex]r = \sqrt[4]{65}[/tex]
Ok, since r does not equal 1, we are not working with a unit circle but we have found r, which is our radius.
Now that we know the value of r, which is [tex]r = 65^{\frac{1}{4}}}[/tex], we need to look at the identities of cos, csc and tan.
The identities:
[tex]cos \theta = \frac{x}{r}[/tex]
[tex]csc \theta = \frac{1}{y}[/tex]
[tex]tan \theta = \frac{y}{x}[/tex]
Now that we know their identities and know the radius of our circle, we can find the exact values of cos, csc and tan.
[tex]cos \theta = \frac{x}{r} = \frac{4}{65^{\frac{1}{4}}}[/tex]
[tex]csc \theta = \frac{1}{y} = \frac{1}{-7}[/tex]
[tex]tan \theta = \frac{y}{x} = \frac{-7}{4}[/tex]
The exact values for cos, csc and tan given the point (4,-7) are:
[tex]cos \theta = \frac{4}{65^{\frac{1}{4}}}[/tex]
[tex]csc \theta = \frac{1}{-7}[/tex]
[tex]tan \theta = \frac{-7}{4}[/tex]
The exact values are [tex]\cos \theta = \frac{4\sqrt{65}}{65}[/tex], [tex]\csc \theta = \frac{\sqrt{65}}{4}[/tex] and [tex]\tan \theta = -\frac{7}{4}[/tex], respectively.
Let be a Point of the form [tex](x,y)[/tex] that is the Terminal Side of angle [tex]\theta[/tex], in sexagesimal degrees. By Trigonometry, we know that the Cosine, Cosecant and Tangent of the Angle are, respectively:
[tex]\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}[/tex] (1)
[tex]\csc \theta = \frac{\sqrt{x^{2}+y^{2}}}{y}[/tex] (2)
[tex]\tan \theta = \frac{y}{x}[/tex] (3)
If we know that [tex]x = 4[/tex] and [tex]y = -7[/tex], then the exact value of all the trigonometric relations are:
[tex]\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}[/tex]
[tex]\cos \theta = \frac{4}{\sqrt{4^{2}+(-7)^{2}}}[/tex]
[tex]\cos \theta = \frac{4\sqrt{65}}{65}[/tex]
[tex]\csc \theta = \frac{\sqrt{x^{2}+y^{2}}}{y}[/tex]
[tex]\csc\theta = \frac{\sqrt{4^{2}+(-7)^{2}}}{4}[/tex]
[tex]\csc \theta = \frac{\sqrt{65}}{4}[/tex]
[tex]\tan \theta = \frac{y}{x}[/tex]
[tex]\tan \theta = -\frac{7}{4}[/tex]
The exact values are [tex]\cos \theta = \frac{4\sqrt{65}}{65}[/tex], [tex]\csc \theta = \frac{\sqrt{65}}{4}[/tex] and [tex]\tan \theta = -\frac{7}{4}[/tex], respectively.
Please see the related question for further details: https://brainly.com/question/15225546
