𝑃(𝜃) lies on the line segment joining the origin and (−8,15). Find the value of 𝑐𝑠𝑐𝜃 + 𝑠𝑒𝑐𝜃.

Answer 1

Hello Coco,

You're given a point P in the form P(θ) = (x,y) where x = -8 and y = 15. x and y represent two of the 3 sides of a right triangle:

Before you go further: find r, the hypotenuse of this right triangle: r = √(-8)² + 15² = √(64+225) = √289 = 17

Knowing all three sides of the right triangle, you can use the definitions of csc and sec to find them separately:

1. csc θ = r/y = 17/15
2. sec θ = r/x = 17/-8 = -17/8

So you got: csc θ = 17/15 and sec θ = -17/8

Last step in answering the problem is adding them up: csc θ + secθ = (17/15) + (-17/8) = (17/15) - (17/8)

Those 2 fractions don't have the same denominator, so you'd want to find a common denominator, then do the math. I'll leave that step up to you, it shouldn't take you too long. You got this!

Answer 2

First, you need to find θ: tanθ = y_p/x_p (through origin) θ = tan^-1(-15/8) which is less than 0 (4th quadrant). Remember that inverse tangent will only return a value between -π/2 and π/2 rads. You must add π to θ to find the solution that is in the second quadrant.

Now plug these two into the equation to calculate cscθ + secθ

If you want an exact solution, then you must make the right triangle formed by the two axes and the line segment to point P , find the hypotenuse and use the sides to determine of the triangle to find csc and sec (you have to put in the right signs for the 2nd quadrant). It's easier to work with 180-theta, the angle of the triangle to the - x axis.