Find the values of the trigonometric functions of t from the given information.
tan(t) =1/9 terminal point of t is in Quadrant III
sin(t)=
cos(t)=
csc(t)=
sec(t)=
cot(t)=
Oh, the terminal point of t is in Quadrant III? That's a tricky spot for trigonometric functions to be in. But fear not, I'm here to help with a little humor!
Let's start with sin(t). Since t is in Quadrant III, the sine function is negative there. Now, imagine a sad sine function: *sin(t) = - the value we're looking for*.
For cos(t), also in Quadrant III, the cosine function is also negative. Picture a gloomy cosine function: *cos(t) = - the value we're looking for*.
Moving on to csc(t), it's the reciprocal of sin(t). And since sin(t) is negative in Quadrant III, csc(t) is the inverse of that negative value. Imagine a grumpy csc(t) who does the opposite of sin(t): *csc(t) = -1/sin(t) = the reciprocal of the value we're looking for* (negative, of course).
Similarly, sec(t) is the reciprocal of cos(t). So, just like csc(t), sec(t) is going to be negative too. It's like a bad influence: *sec(t) = -1/cos(t) = the reciprocal of the value we're looking for* (negative, as always).
Finally, we have cot(t), which is the reciprocal of tan(t). Remember that sad little tan(t) in Quadrant III? Well, the cot(t) is its opposite! Imagine a cheerful cot(t) that does the opposite of tan(t): *cot(t) = 1/tan(t) = the reciprocal of the value we're looking for* (but positive this time!).
So, to summarize:
sin(t) = - the value we're looking for
cos(t) = - the value we're looking for
csc(t) = -1/reciprocal of the value we're looking for
sec(t) = -1/reciprocal of the value we're looking for
cot(t) = reciprocal of the value we're looking for
Hope that brightened up the world of trigonometry for you!
Given that tan(t) = 1/9 and the terminal point of t is in Quadrant III, we can find the values of the other trigonometric functions as follows:
Since tan(t) = 1/9, we can determine that sin(t) = 1/sqrt(1 + tan(t)^2) and cos(t) = tan(t) * cos(t).
To find sin(t), we substitute the value of tan(t) into the formula:
sin(t) = 1/sqrt(1 + (1/9)^2)
sin(t) = 1/sqrt(1 + 1/81)
sin(t) = 1/sqrt(82/81)
sin(t) = sqrt(81/82)
sin(t) = sqrt(81)/sqrt(82)
sin(t) = 9/sqrt(82)
To find cos(t), we multiply tan(t) by cos(t):
cos(t) = tan(t) * cos(t) = 1/9 * cos(t)
To find csc(t) (cosecant), we take the reciprocal of sin(t):
csc(t) = 1/sin(t) = 1/(9/sqrt(82))
csc(t) = sqrt(82)/9
To find sec(t) (secant), we take the reciprocal of cos(t):
sec(t) = 1/cos(t) = 1/(1/9 * cos(t))
sec(t) = 9/cos(t)
To find cot(t) (cotangent), we take the reciprocal of tan(t):
cot(t) = 1/tan(t) = 1/(1/9)
cot(t) = 9
Therefore, the values of the trigonometric functions are:
sin(t) = 9/sqrt(82)
cos(t) = 1/9 * cos(t)
csc(t) = sqrt(82)/9
sec(t) = 9/cos(t)
cot(t) = 9
To find the values of the trigonometric functions of t, we need to use the given information that tan(t) = 1/9 and that the terminal point of t is in Quadrant III.
First, let's identify which trigonometric functions are necessary to find the values requested:
1. sine (sin): To find sin(t), we need to determine the value of the y-coordinate in the Cartesian coordinate system.
2. cosine (cos): To find cos(t), we need to determine the value of the x-coordinate in the Cartesian coordinate system.
3. cosecant (csc): To find csc(t), we need to calculate the reciprocal of sin(t).
4. secant (sec): To find sec(t), we need to calculate the reciprocal of cos(t).
5. cotangent (cot): To find cot(t), we need to calculate the reciprocal of tan(t).
To find sin(t), cos(t), csc(t), sec(t), and cot(t) for the given information, we can follow these steps:
Step 1: Find the value of tan(t): tan(t) = 1/9.
Since the terminal point of t is in Quadrant III (where both the x and y-coordinates are negative), we know that the value of tan(t) will also be negative.
Step 2: Determine the value of tan(t) in Quadrant III.
We can use the equation tan(t) = opposite/adjacent = y/x.
Given that tan(t) = 1/9, we can let y = -1 and x = -9.
Step 3: Determine the values of sin(t) and csc(t):
Since sin(t) = y/r, where r is the radius of the unit circle (which is always positive), we can use the Pythagorean theorem to find r:
r^2 = x^2 + y^2
r^2 = (-9)^2 + (-1)^2 = 82
r = √82
Therefore, sin(t) = y/r = -1/√82.
And csc(t) = 1/sin(t) = -√82.
Step 4: Determine the values of cos(t) and sec(t):
Since cos(t) = x/r, we can find cos(t) using the values from Step 2 and the value of r calculated in Step 3:
cos(t) = x/r = -9/√82.
And sec(t) = 1/cos(t) = -√82/9.
Step 5: Determine the value of cot(t):
Since cot(t) = 1/tan(t) = 1/(1/9) = 9.
Therefore, the values of the trigonometric functions for t are:
sin(t) = -1/√82,
cos(t) = -9/√82,
csc(t) = -√82,
sec(t) = -√82/9,
cot(t) = 9.
In QIII you have
x = -9
y = -1
r = √82
So, now just recall the definitions of the trig functions for a triangle in standard position.