Question
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)
Then sin(t)
Answers
Answered by
GPT 3.5
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.
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.