To find sin(t),cos(t)and tan(t) for the given terminal point p(x,y)=(-2/5,21/5) we can use the definition of these trignometric functions in relation to the coordinates of the terminal point on the unit circle
To find sin(t), cos(t), and tan(t) for the given terminal point P(x,y) = (-2/5, 21/5), we can use the definition of these trigonometric functions in relation to the coordinates of the terminal point on the unit circle.
Let r be the radius of the unit circle, which is always 1.
Using the Pythagorean theorem, we can find the value of r by:
r^2 = x^2 + y^2
r^2 = (-2/5)^2 + (21/5)^2
r^2 = 4/25 + 441/25
r^2 = 445/25
r = sqrt(445)/5
Now, we can find sin(t) and cos(t) using the coordinates x and y:
sin(t) = y / r
sin(t) = (21/5) / (sqrt(445)/5)
sin(t) = 21 / sqrt(445)
cos(t) = x / r
cos(t) = (-2/5) / (sqrt(445)/5)
cos(t) = -2 / sqrt(445)
Finally, we can find tan(t) using the definition:
tan(t) = sin(t) / cos(t)
tan(t) = (21 / sqrt(445)) / (-2 / sqrt(445))
tan(t) = -21 / 2
Therefore, for the given terminal point P(x,y) = (-2/5, 21/5), we have:
sin(t) = 21 / sqrt(445)
cos(t) = -2 / sqrt(445)
tan(t) = -21 / 2
To find sin(t), cos(t), and tan(t) for the given terminal point p(x, y) = (-2/5, 21/5), we can use the definition of these trigonometric functions in relation to the coordinates of the terminal point on the unit circle.
Let's start by finding the angle t:
We can use the inverse tangent function (arctan) to find the angle t. Therefore, t = arctan(y/x).
Given: x = -2/5 and y = 21/5
t = arctan((21/5)/(-2/5))
t = arctan(-21/2)
Next, we can find sin(t), cos(t), and tan(t) using the sine, cosine, and tangent functions:
sin(t) = y / r
cos(t) = x / r
tan(t) = y / x
Where r is the radius of the unit circle, which is always 1.
Let's substitute the values of x, y, and t to find the trigonometric functions:
r = 1
sin(t) = (21/5) / 1 = 21/5
cos(t) = (-2/5) / 1 = -2/5
tan(t) = (21/5) / (-2/5) = -21/2
Therefore, for the given terminal point p(x, y) = (-2/5, 21/5), the trigonometric values are:
sin(t) = 21/5
cos(t) = -2/5
tan(t) = -21/2
To find sin(t), cos(t), and tan(t) for the given terminal point P(x, y) = (-2/5, 21/5), we can use the definition of these trigonometric functions in relation to the coordinates of the terminal point on the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin (0, 0). It is commonly used in trigonometry to define the trigonometric functions for any angle.
To find sin(t), cos(t), and tan(t), we need to determine the angle t that corresponds to the given terminal point P(x, y) on the unit circle.
First, we can find the hypotenuse of the right triangle formed by the terminal point P(x, y) and the x-axis. The hypotenuse of this triangle is the radius of the unit circle, which is always 1.
Since P(x, y) = (-2/5, 21/5), the x-coordinate represents the adjacent side of the triangle, and the y-coordinate represents the opposite side of the triangle.
Using the Pythagorean theorem, we can determine the length of the hypotenuse:
hypotenuse^2 = adjacent^2 + opposite^2
1^2 = (-2/5)^2 + (21/5)^2
1 = 4/25 + 441/25
1 = 445/25
Now, we can find the angle t by using the inverse trigonometric function:
t = arctan(opposite/adjacent) = arctan((21/5) / (-2/5))
t = arctan(-21/2)
t ≈ -2.819 (radians)
Now that we have the angle t, we can find sin(t), cos(t), and tan(t):
sin(t) = y-coordinate/hypotenuse = (21/5) / 1 = 21/5
cos(t) = x-coordinate/hypotenuse = (-2/5) / 1 = -2/5
tan(t) = sin(t)/cos(t) = (21/5) / (-2/5) = -21/2