Find the exact value of indicated trignometric function of theta.
Sin= -4/9 , tan theta>0 find sec
Since the sine is negative and the tangent is positive, the angle is in quadrant III
from sin theta = -4/9, r = 9 and y = -4
x^2 + y^2 = r^2
x^2 + 16 = 81
x = -sqrt(65) , in III the x is negativ
Then cos theta = -sqrt(65)/9
sec theta = -9/sqrt(65)
or -9sqrt(65)/65
To find sec(theta), we need to use the Pythagorean identity for secant:
sec(theta) = 1 / cos(theta)
Since the tangent of theta is positive (tan(theta) > 0), we can determine the sign of cosine using the information from the tangent function using the quadrants of the unit circle.
In quadrant II and quadrant IV, both sine and tangent are positive. Since we know the sine is negative (-4/9), theta must be in quadrant IV.
In quadrant IV, sine is negative, and cosine is positive.
To find the cosine of theta, we can use the Pythagorean identity for sine:
sin^2(theta) + cos^2(theta) = 1
(-4/9)^2 + cos^2(theta) = 1
16/81 + cos^2(theta) = 1
cos^2(theta) = 1 - 16/81
cos^2(theta) = (81 - 16)/81
cos^2(theta) = 65/81
Since cosine is positive in quadrant IV, taking the square root of both sides, we have:
cos(theta) = sqrt(65/81)
cos(theta) = sqrt(65) / sqrt(81)
cos(theta) = sqrt(65) / 9
Now that we have the cosine, we can find sec(theta):
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (sqrt(65) / 9)
sec(theta) = 9 / sqrt(65)
Therefore, the exact value of sec(theta) is 9/sqrt(65).
To find the value of secant (sec), we need to determine the reciprocal of the cosine function.
Here's how you can find the exact value of sec(theta):
1. Start with the given information that sin(theta) = -4/9 and tan(theta) > 0.
2. Since sin(theta) = -4/9, we can determine that the hypotenuse is 9 and the opposite side is -4 in a right triangle.
3. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this, we can find the length of the adjacent side.
Using the Pythagorean theorem: 9^2 = (-4)^2 + adjacent side^2
Simplifying: 81 = 16 + adjacent side^2
Solving for adjacent side^2: adjacent side^2 = 81 - 16 = 65
Taking the square root: adjacent side = √65
4. Now, we have the values for the opposite side (-4), hypotenuse (9), and adjacent side (√65). We can find the cosine of theta.
cosine(theta) = adjacent side / hypotenuse
cosine(theta) = √65 / 9
5. Finally, we can find the value of sec(theta) by taking the reciprocal of cosine(theta).
sec(theta) = 1 / cosine(theta)
sec(theta) = 1 / (√65 / 9)
sec(theta) = 9 / √65
Therefore, the exact value of sec(theta) is 9 / √65.