Find the exact value, given the conditions Sin(alpha) = 3/5 , 0< alpha< pi/2, (a) sin( alpha + beta)

Question

In my book, it says the answer is 2√5/ 25, if you use a formula please write it out so I know how you got certain answers. Use the special triangle with the sides of 3-4-5, please and thank you!

Answer 1

So, we start with Sin(α)=3/5. Going back to right triangle trig, remeber that Sin is defined as opposite/hypotenuse. Therefore, for the angle α, The sin of this angle is opposite/hypotenuse = 3/5. So, the opposite side of angle α is 3, and the hypotenuse is 5. From right triangle trig, we know also now that the adjacent side to angle α must be 4, completing the pythagorean 3-4-5 triple.

As I cannot draw a picture here, I will do my best to explain how I'm understanding the triangle to look

There is a right triangle, with the larger angle β on the top, the smaller angle α on the bottom, and a right angle. The hypotenuse of the triangle has length 5, while the height of the triangle aka the side opposite angle α has length 3. The side adjacent to angle α aka the side opposite angle β has length 4.

Given this, we can now write down Sin, Cos, and Tan for each angle, keeping in mind SOHCAHTOA

Sin(α)=opposite/hypotenuse=3/5

Cos(α)=adjacent/hypotenuse=4/5

Tan(α)=opposite/adjacent=3/4

Sin(β)=side opposite angle β/hypotenuse=4/5

Cos(β)=side adjacent angle β/hypotenuse=3/5

Tan(β)=side opposite β/side adjacent β = 4/3

I have written all of this out because there is a formula for Sin(α+β), which should be in your trigonometry book. The addition formula for Sin is

Sin(α+β)=Sin(α)Cos(β)+Cos(α)Sin(β)

Now, we just plug this in using what we have above

Sin(α+β)=(3/5)(3/5)+(4/5)(4/5)=9/25+16/25=25/25=1

The reason you get 1 in this instance is because, in a right triangle, adding the two angles which are not the right angle will give you 90 degrees, as 90+90=180 and all triangles add up to 180 degrees, and the sin of 90 degrees is 1.

Is it possible to give more context for the problem please? I'd love to continue working with you if I had more information on the problem to see where the answers are going different. Good luck!