Suppose tan(theta)=5/7 and theta belongs to Quadrant III, find the exact value of sec(theta)?

I already took the steps to solve this problem but I am confused if whether or not sec(theta)is positive or negative sqrt(74)/7

Any help is greatly appreciated!

Make a sketch in your notes regarding CAST

It tells you in which quadrants the main trig ratios of (S)ine, (C)osine and (T)angent are positive.
Of course if the cosine is positive then its reciprocal, the secant, is also positive, etc

since tanØ = 5/7 = y/x
x = -7 , y = -5 , since we are in quadrant III
r^2 = 7^2 + 5^2
r = √74

cosØ = x/r = -7/√74
secØ = -√74/7

The CAST rule makes all this +/- stuff really easy.

Got it, thanks for the help!

To find the exact value of sec(theta), we need to recall the reciprocal identities of trigonometric functions. The reciprocal of tangent is secant, so we can rewrite the equation using the reciprocal identity:

sec(theta) = 1/cos(theta)

To determine whether sec(theta) is positive or negative, we can use the information that theta belongs to Quadrant III. In Quadrant III, both x and y coordinates are negative.

Since tan(theta) = 5/7 and theta belongs to Quadrant III, we can determine the remaining side lengths of the triangle in Quadrant III using the Pythagorean theorem:

Let's assume that the opposite side is 5k and the adjacent side is 7k.

Then, using the Pythagorean theorem, we have:

(7k)^2 = (5k)^2 + (x)^2
49k^2 = 25k^2 + x^2
24k^2 = x^2

Taking the square root of both sides, we get:

sqrt(24k^2) = sqrt(x^2)
2√6k = √x

Since cos(theta) is adjacent/hypotenuse, we can express cos(theta) as:

cos(theta) = adjacent/hypotenuse = (7k)/hypotenuse

Using the Pythagorean theorem, the hypotenuse is:

hypotenuse = √[(7k)^2 + (5k)^2]
= √(49k^2 + 25k^2)
= √(74k^2)
= √74k

Substituting the values into cos(theta), we have:

cos(theta) = (7k)/(√74k)

Now we can substitute cos(theta) into sec(theta) using the reciprocal identity:

sec(theta) = 1/(cos(theta))
= 1/[(7k)/(√74k)]
= √74k/(7k)

The k's cancel out:

sec(theta) = √74/7

From this, we can conclude that sec(theta) is positive, since we have taken the absolute value of the expression. Therefore, the exact value of sec(theta) when tan(theta) = 5/7 and theta belongs to Quadrant III is √74/7.