for the point deterime all six trigonometric ratios and the value of the principal angle and the related acute angle
P( -3,0)
I assume the triangle has the line OP as the hypotenuse.
In this case, the point corresponds to an angle of 180°, so just go from there.
To determine the trigonometric ratios for the point P(-3,0), we can first find the value of the principal angle using the coordinates of the point. Then, we can compute the trigonometric ratios based on that angle.
1. Principal Angle (θ):
To find the principal angle, we can use the inverse tangent function, which is commonly denoted as atan or tan^(-1). The formula to calculate the principal angle is:
θ = atan(y / x)
Using the coordinates of P(-3,0), we substitute x = -3 and y = 0 into the formula:
θ = atan(0 / -3)
θ = atan(0)
θ = 0 degrees (or 0 radians)
Therefore, the principal angle is 0 degrees (or 0 radians).
2. Related Acute Angle:
The related acute angle is simply the positive form of the principal angle. In this case, since the principal angle is 0 degrees, the related acute angle is also 0 degrees.
3. Trigonometric Ratios:
Now, we can calculate the six trigonometric ratios using the principal angle (θ) and related acute angle.
- Sine (sin):
sin(θ) = sin(0) = 0
sin(related acute angle) = sin(0) = 0
- Cosine (cos):
cos(θ) = cos(0) = 1
cos(related acute angle) = cos(0) = 1
- Tangent (tan):
tan(θ) = tan(0) = 0
tan(related acute angle) = tan(0) = 0
- Cosecant (csc):
csc(θ) = 1 / sin(θ) = 1 / 0 (undefined)
csc(related acute angle) = 1 / sin(0) = 1 / 0 (undefined)
- Secant (sec):
sec(θ) = 1 / cos(θ) = 1 / 1 = 1
sec(related acute angle) = 1 / cos(0) = 1 / 1 = 1
- Cotangent (cot):
cot(θ) = 1 / tan(θ) = 1 / 0 (undefined)
cot(related acute angle) = 1 / tan(0) = 1 / 0 (undefined)
Therefore, the trigonometric ratios for the point P(-3,0) are as follows:
sin = 0
cos = 1
tan = 0
csc = undefined
sec = 1
cot = undefined