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5cosA+3=0 and180<A<360, without using a calculator and with the aid of diagram, determine the value of SinA+ cosA

A is in QIII or QIV

cosA = -3/5 so A is in QIII, and
x = -3
r = 5
y = -4
sinA = y/r = -4/5
sinA + cosA = -4/5 - 3/5 = -7/5

Did you know?

Did you know that in trigonometry, you can determine the value of sin A + cos A without using a calculator? Given the equation 5cos A + 3 = 0 and the condition 180 < A < 360, we can first solve for cos A by isolating the term. Subtracted 3 from both sides, it becomes 5cos A = -3. Next, dividing both sides by 5, we find cos A = -3/5. Now, looking at the given range, we know that A is an angle in the third quadrant. In the third quadrant, both sin A and cos A are negative. Since we know the value of cos A, we can determine sin A by utilizing the Pythagorean identity sin^2 A + cos^2 A = 1. Plugging in the value of cos A, we get sin^2 A + (-3/5)^2 = 1. Simplifying the equation, we have sin^2 A + 9/25 = 1. Rearranging the equation, sin^2 A = 1 - 9/25, which is sin^2 A = 16/25. Taking the square root of both sides, we find sin A = +/- 4/5. However, since A is in the third quadrant, sin A is negative. Therefore, sin A = -4/5. Finally, we can determine the value of sin A + cos A by simply adding the values we obtained. sin A + cos A = (-4/5) + (-3/5) = -7/5.