Given that sin A=5/3 and A is obtuse, find the value of the following without using calculator

Question

A) 2cosA+3sinA (b) 4tanA-2cosA

Answer 1

Sin=opposite/hypotenuse.

Since the larger # is the numerator; something's wrong.

Answer 2

It is not possible for the sin(A) to be greater than 1 so 5/3 is not possible. So either the answer is "no solution" or you typed something in error. I'm going to assume that you did a typo and you meant to say sin(A) = 3/5

Based on that assumption:

Remember that sin(A) = opposite/hypotenuse so when we are told that If sin(A) = 3/5 then we are being told that the opposite side is 3 and the hypotenuse is 5. If we draw out the triangle, and put it on a coordinate plane (x-y axis), the triangle would look like this:

however, you will notice that angle A is not obtuse. So this isn't quite correct. What if we flip it around so that angle A is in quadrant 2 like this:

It makes the blue triangle but the angle we are considering is actually shown in red as angle A.

Using the blue triangle we can use the Pythagorean Theorem to solve for the missing side:

Notice that we listed the missing side as negative (-4), the Pythagorean Theorem tells us the side is 4, but when we consider the coordinate plane, the value of the "x" coordinate must be negative for that obtuse angle.

Now we can write all the other trig ratios.

cos(A) = -4/5

tan(A) = 3/-4

So:

2cos(A) + 3sin(A) = 2(-4/5) + 3(3/5) = -8/5 + 9/5 = 1/5

and

4tan(A) - 2cos(A) = 4(3/-4) - 2(-4/5) = -3 - -8/5 = -15/5 + 8/5 = -7/5