Find the values of the trigonmeric functions of t from the given information
cos t = -4/5 terminal point of t is in quadrant III
First find the absolute values of each function, and assign the sign accordingly for quadrant 3.
The signs of the different functions are as follows:
S|A
---
T|C
which shows that in Quad 1, All trigonometric functions are positive, in Q2, only Sine and cosecant are positive. In Q3, Tangent and cotangent are positive. In Q4, Cosine and secant are positive.
cos(t)=-4/5,
sin(t)=√(1-(-4/5)²)
=3/5
The sign for sin(t) in quadrant 3 is -ve, so sin(t)=-3/5.
tan(t)=sin(t)/cos(t)=3/4
In Q3, tangents are positive, so tan(t)=3/4.
I'll leave it to you to figure out the remaining functions, namely cosecant, secant and cotangent.
Well, if cos t = -4/5 and the terminal point of t is in quadrant III, then we can use the Pythagorean identity to find the value of sin t.
We know that cos t = adj/hyp = -4/5, so we can let adj = -4 and hyp = 5.
Using the Pythagorean theorem, we can find the value of the opposite side (opp):
opp = √(hyp^2 - adj^2) = √(5^2 - (-4)^2) = √(25 - 16) = √9 = 3.
So, sin t = opp/hyp = 3/5.
Now, let's find the values of the other trigonometric functions:
tan t = sin t / cos t = (3/5) / (-4/5) = -3/4.
Sec t = 1 / cos t = 1 / (-4/5) = -5/4.
Cosec t = 1 / sin t = 1 / (3/5) = 5/3.
Cot t = 1 / tan t = 1 / (-3/4) = -4/3.
So, the values of the trigonometric functions for t are:
sin t = 3/5, cos t = -4/5, tan t = -3/4, sec t = -5/4, cosec t = 5/3, cot t = -4/3.
Hope that helps! And remember, when it comes to trigonometry, always find a way to have a sine of humor!
To find the values of the trigonometric functions of t, we will use the information given that cos t = -4/5 and the terminal point of t is in quadrant III.
In quadrant III, the x-coordinate is negative and the y-coordinate is negative.
Since cos t = adjacent/hypotenuse, we can let the adjacent side of the angle t be -4 (negative because it lies in quadrant III) and the hypotenuse be 5 (from the given information).
Using the Pythagorean theorem, we can find the opposite side of the triangle:
opposite^2 = hypotenuse^2 - adjacent^2
opposite^2 = 5^2 - (-4)^2
opposite^2 = 25 - 16
opposite^2 = 9
opposite = √9
opposite = 3
Now, we have the values of the adjacent side (adjacent = -4), opposite side (opposite = 3), and hypotenuse (hypotenuse = 5).
Using these values, we can find the values of the trigonometric functions:
sin t = opposite/hypotenuse = 3/5
cos t = adjacent/hypotenuse = -4/5 (given)
tan t = opposite/adjacent = 3/-4 = -3/4
csc t = 1/sin t = 1/(3/5) = 5/3
sec t = 1/cos t = 1/(-4/5) = -5/4
cot t = 1/tan t = 1/(-3/4) = -4/3
So, the values of the trigonometric functions of t are as follows:
sin t = 3/5
cos t = -4/5
tan t = -3/4
csc t = 5/3
sec t = -5/4
cot t = -4/3
To find the values of the trigonometric functions of t, we use the given information that cos t is -4/5 and the terminal point of t is in quadrant III.
In quadrant III, the x-coordinate is negative and the y-coordinate is negative. Therefore, we can write:
cos t = x-coordinate / hypotenuse
Since cos t is given as -4/5, we can substitute this value into the equation:
-4/5 = x / hypotenuse
To find the value of x, we can use the Pythagorean identity, which states:
cos^2 t + sin^2 t = 1
Substituting the given value of cos t:
(-4/5)^2 + sin^2 t = 1
16/25 + sin^2 t = 1
sin^2 t = 1 - 16/25
sin^2 t = 9/25
Taking the square root of both sides, we find:
sin t = ±√(9/25)
sin t = ±3/5
Since t is in quadrant III, sin t is negative. Therefore, we take the negative value:
sin t = -3/5
Now we can use this information to find the remaining trigonometric functions.
tan t = sin t / cos t
= (-3/5) / (-4/5)
= 3/4
cot t = 1 / tan t
= 1 / (3/4)
= 4/3
sec t = 1 / cos t
= 1 / (-4/5)
= -5/4
csc t = 1 / sin t
= 1 / (-3/5)
= -5/3
Therefore, the values of the trigonometric functions of t are:
sin t = -3/5
cos t = -4/5
tan t = 3/4
cot t = 4/3
sec t = -5/4
csc t = -5/3