Find the values of the trigonometric function of t from the given information sec(t)=2 terminal point of t is in quadrant 4
Then sin(t)
To find the value of sin(t), we can use the given information that sec(t) = 2.
Recall that sec(t) is the reciprocal of cosine. So, if sec(t) = 2, then cosine(t) = 1/2.
Since the terminal point of t is in quadrant 4, the x-coordinate of the terminal point is positive and the y-coordinate is negative.
In quadrant 4, the reference angle is formed by drawing a line from the terminal point to the x-axis.
Since cosine is positive in quadrant 4, and we know that cosine(t) = 1/2, we can deduce that the reference angle (let's call it A) has a cosine of 1/2.
So, cos(A) = 1/2.
Now, we need to find the value of sin(A).
To find sin(A), we can use the Pythagorean identity: sin^2(A) + cos^2(A) = 1.
Substituting the value of cos(A) = 1/2, we get: sin^2(A) + (1/2)^2 = 1.
Simplifying, we have: sin^2(A) + 1/4 = 1.
Subtracting 1/4 from both sides, we get: sin^2(A) = 3/4.
Taking the square root of both sides, we have: sin(A) = sqrt(3)/2.
Therefore, sin(t) = sin(A) = sqrt(3)/2.
To find the value of sin(t), we can use the relationship between the trigonometric functions in a right triangle.
Since sec(t) = 2 and the terminal point of t is in quadrant 4, we know that sec(t) is positive in quadrant 4. In a right triangle, sec(t) is the ratio of the hypotenuse to the adjacent side.
Let's assume that the adjacent side of the triangle is a, and the hypotenuse is h.
We can use the Pythagorean theorem to find the length of the opposite side (o) of the triangle:
a^2 + o^2 = h^2
Since sec(t) is the ratio of h to a, we know that h = 2a.
Substituting this into the Pythagorean theorem equation, we get:
a^2 + o^2 = (2a)^2
a^2 + o^2 = 4a^2
3a^2 = o^2
Taking the square root of both sides, we get:
sqrt(3)a = o
Now, we can use the trigonometric identity sin(t) = o/h to find the value of sin(t):
sin(t) = (sqrt(3)a) / (2a)
Simplifying the expression, we get:
sin(t) = sqrt(3) / 2
Therefore, the value of sin(t) when sec(t) = 2 and the terminal point of t is in quadrant 4 is sqrt(3) / 2.
To find the value of sin(t) given that sec(t) = 2 and the terminal point of t is in quadrant 4, we can use the Pythagorean identity sin^2(t) + cos^2(t) = 1.
Since sec(t) = 2, we know that the adjacent side to t is 2 units long and the hypotenuse is 1 unit long. From this information, we can determine that the opposite side must be sqrt(1^2 - 2^2) = sqrt(-3).
However, since the terminal point of t is in quadrant 4, which is the right-bottom quadrant, the value of sin(t) should be positive. Therefore, the square root of a negative number is not possible in this case.
Hence, there are no values of t that satisfy sec(t) = 2 and the terminal point of t being in quadrant 4, and therefore sin(t) cannot be determined.