The terminal side of θ lies on a given line in the specified quadrant. Find the values of the six trigonometric functions of θ by finding a point on the line. Line: (55/48)x Quadrant:III
sin(θ)=
cos(θ)=
tan(θ)=
csc(θ)=
sec(θ)=
cot(θ)=
Line: 4x+3y=0 IV
To find the values of the six trigonometric functions of θ, we need to find a point on the line in the specified quadrant.
Given line: (55/48)x
Quadrant: III
In the third quadrant, both the x and y coordinates are negative.
To find a point on the line, let's assume x = -48, and we can calculate the corresponding y value.
Substituting x = -48 into the equation of the line:
y = (55/48)(-48)
y = -55
So, a point on the line in the third quadrant is (-48, -55).
Now, let's calculate the trigonometric functions of θ based on this point.
sin(θ) = y/r = -55/√((-48)^2 + (-55)^2)
cos(θ) = x/r = -48/√((-48)^2 + (-55)^2)
tan(θ) = y/x = -55/(-48)
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)
Now, let's calculate the values:
sin(θ) = -55/√(48^2 + 55^2)
cos(θ) = -48/√(48^2 + 55^2)
tan(θ) = -55/-48
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)
To find the values of the six trigonometric functions of θ, we need to find a point on the given line in the specified quadrant (QIII).
The equation of the line is given by (55/48)x. In QIII, the x-values are negative, and y-values are also negative.
Let's choose an x-value that is negative and substitute it into the equation to find the corresponding y-value.
Considering x = -48 (a negative value since we're in QIII), we can calculate the y-coordinate as follows:
y = (55/48)(-48)
= -55
So, the point on the line in QIII is (-48, -55).
Now, let's calculate the six trigonometric functions using this point:
1. sin(θ) = y-coordinate / hypotenuse = -55 / hypotenuse
To find the hypotenuse, we can use the distance formula or Pythagorean theorem. Since the point lies on the line, the hypotenuse can be calculated as follows:
hypotenuse = √(x-coordinate² + y-coordinate²)
= √((-48)² + (-55)²)
= √(2304 + 3025)
= √5329
= 73
Therefore, sin(θ) = -55 / 73.
2. cos(θ) = x-coordinate / hypotenuse = -48 / hypotenuse
cos(θ) = -48 / 73
3. tan(θ) = sin(θ) / cos(θ)
tan(θ) = (-55 / 73) / (-48 / 73)
tan(θ) = -55 / -48
tan(θ) = 55 / 48
4. csc(θ) = 1 / sin(θ)
csc(θ) = 1 / (-55 / 73)
csc(θ) = 73 / -55
5. sec(θ) = 1 / cos(θ)
sec(θ) = 1 / (-48 / 73)
sec(θ) = 73 / -48
6. cot(θ) = 1 / tan(θ)
cot(θ) = 1 / (55 / 48)
cot(θ) = 48 / 55
Therefore, the values of the six trigonometric functions of θ are as follows:
sin(θ) = -55 / 73
cos(θ) = -48 / 73
tan(θ) = 55 / 48
csc(θ) = 73 / -55
sec(θ) = 73 / -48
cot(θ) = 48 / 55
I will assume you are saying that the line is
y = (55/48)x
or one point on the line in quadrant III would be
(-48,-55)
then r^2 = (-48)^2 + (-55)^2
r = √5329 = 73
sinØ = y/r = -55/73
cosØ = x/r = -48/73
etc.