The statement "tane = -1/2/2, e is in quadrant 3": CSC8 = 13, and the terminal point determined by theta is in quadrant 3
The terminal point determined by theta is (13, -1).
-1/2/2 is not properly defined, is it
-1 / (2/2) or is it (-1/2) / 2 , I will assume the latter ,
so we would have tan e = -1/4 , where e is in III
the rest of the question makes no sense to me
if csc 8 = 13, then sin 8 = 1/13 , angle names are usually not given digit names.
where does theta enter the picture???
To determine the value of "tane" and the quadrant of angle "e," we will use the given information.
CSC8 = 13 means that the cosecant of angle 8 is equal to 13. The cosecant of an angle is equal to the reciprocal of the sine of that angle.
Cosecant is defined as: CSC(theta) = 1/sin(theta)
Therefore, 1/sin8 = 13
To solve this equation, we need to find the value of sin8. We can use the reciprocal identity of sine to find sin8.
Reciprocal identity of sine: sin(theta) = 1/csc(theta)
sin8 = 1/13
Now, we can find the value of "tane" using the formula tangent = sine/cosine.
tane = sin8/cosine8
To find the value of cosine8, we can use the identity: cos(theta) = 1/sce(theta).
cos8 = 1/13
Therefore, tane = sin8/cos8 = (1/13)/(1/13) = 1
Now, we need to determine the quadrant of angle "e". From the given information, we know that the terminal point determined by theta is in quadrant 3. In this quadrant, both the sine and tangent values are negative.
Since tane = 1, we need to negate it to match the information provided:
tane = -1
Therefore, the statement "tane = -1/2/2, e is in quadrant 3" is not valid based on the given information.
To understand this problem, we need to know the definitions of the trigonometric functions and the relationship between the trigonometric functions and the coordinates of terminal points on the unit circle.
In trigonometry, the coordinates of a point on the unit circle can be represented as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis to the terminal side of the angle.
The trigonometric functions are defined as follows:
- Sine (sin θ) is equal to the y-coordinate of the point on the unit circle.
- Cosine (cos θ) is equal to the x-coordinate of the point on the unit circle.
- Tangent (tan θ) is equal to the ratio of sine to cosine, which is y/x.
- Cosecant (csc θ) is equal to the reciprocal of sine, which is 1/sin θ.
Now, let's use the given information to find the value of CSC8.
We are given that the terminal point determined by θ is in quadrant 3. In quadrant 3, both the x-coordinate and the y-coordinate are negative. Since the sine function is equal to the y-coordinate of the terminal point, and the terminal point is in quadrant 3, we can conclude that sin(θ) is negative.
We are also given that tan(θ) = -1/2. Since tan(θ) is equal to the ratio of sine to cosine, we can write the equation tan(θ) = sin(θ) / cos(θ). Substituting the given value, we get (-1/2) = sin(θ) / cos(θ).
To find sin(θ), we can multiply both sides of the equation by cos(θ): (-1/2) * cos(θ) = sin(θ).
Now, we need to find the value of cos(θ) to solve the equation. Since we know that the terminal point is in quadrant 3, the x-coordinate of the point is negative. Cosine (cos θ) is equal to the x-coordinate of the point, so we can conclude that cos(θ) is negative.
Let's assume that cos(θ) is -2, which satisfies the condition that cos(θ) is negative in quadrant 3. We can substitute this value into our equation: (-1/2) * (-2) = sin(θ).
Simplifying the equation, we get sin(θ) = 1.
Now that we have found the values of sin(θ) and cos(θ), we can find the value of csc(θ). Cosecant (csc θ) is equal to the reciprocal of sine, so csc(θ) = 1 / sin(θ).
Substituting the value of sin(θ) we found earlier, we get csc(θ) = 1 / 1 = 1.
Therefore, CSC8 = 13 and the terminal point determined by θ is in quadrant 3.