Find (sinϴ + cosϴ) if the terminal side of ϴ passes through (5,2)
the x=5, y=2 and
r^2 = 25+4 = 29
r = √29
sinØ = 2/√29 + 5/√29 = 7/√29
To find (sinϴ + cosϴ), we need to determine the values of sine (sinϴ) and cosine (cosϴ) of the angle ϴ.
Given that the terminal side of ϴ passes through the point (5, 2), we can use the coordinates of this point to find the values of sine and cosine.
First, let's determine the length of the hypotenuse using the Pythagorean theorem. The hypotenuse is the distance between the origin (0, 0) and the point (5, 2). Applying the theorem:
hypotenuse = √((5)^2 + (2)^2)
= √(25 + 4)
= √29
Next, we need to find the values of sine and cosine. Since the point (5, 2) is in the first quadrant (both x and y coordinates are positive), we can determine the sine and cosine directly using the definitions:
sinϴ = opposite / hypotenuse
= 2 / √29
cosϴ = adjacent / hypotenuse
= 5 / √29
Now, we can substitute these values into the expression (sinϴ + cosϴ):
(sinϴ + cosϴ) = (2 / √29) + (5 / √29)
= (2 + 5) / √29
= 7 / √29
Therefore, (sinϴ + cosϴ) = 7 / √29.