Write sec in terms of sin. Theta is in the third Quadrant.
You mean secØ and sinØ, the words sec and sin by themselves are just mathematical operators like
+, - , √, etc, they can't just stand alone.
anyway.....
secØ = 1/cosØ
but cosØ = ±√(1-sin^2 Ø)
since you state the angle to be in III
secØ = -1/√(1 - sin^2 Ø)
Thank you. I know that sec and sin cannot stand alone, I just don't know how to type the symbol for theta. Could you help me with a similar problem? It involves expressing tan(t) in terms of cos(t), where t is in the third quadrant.
To write sec in terms of sin, we need to use the concept of reciprocal trigonometric functions.
In the third quadrant, the x-coordinate is negative and the y-coordinate is also negative. Let's consider a point (x, y) in the third quadrant, where the hypotenuse of the triangle formed by the point and the origin is 1.
Since theta is in the third quadrant, the sine of theta is negative. Therefore, sin(theta) = -y.
We know that secant (sec) is defined as the reciprocal of cosine (cos). Since cosine is the ratio of the adjacent side to the hypotenuse, in the third quadrant, the adjacent side is negative (x-coordinate) and the hypotenuse is positive (1).
In terms of the given point, we have cos(theta) = x, and sec(theta) = 1/cos(theta).
So, to write sec(theta) in terms of sin(theta) for theta in the third quadrant:
Start with sin(theta) = -y
Then, use Pythagorean theorem: x^2 + y^2 = 1, to find x.
Lastly, substitute x into the equation sec(theta) = 1/cos(theta) to get sec(theta) in terms of sin(theta).
Note: It would be helpful to know the specific value of sin(theta) in order to calculate sec(theta) precisely.