Calculating expressions using sin of radians

[Emerald123]

I don't know whether or not I missed something during my last math class... I forgot my glasses on that day.

I don't understand what exactly I'm doing wrong while trying to calculate these two expressions:

1. 3 sin pi/2 - 2 sin 5pi/4 + 1/2 sin pi
For this one I begun by converting a few sines: 3- 2 sin 5pi/4 + 0. Am I doing something completely wrong? And if what I did so far is right, how do I calculate 2 sin 5pi/4?
According to my book, the answer is 3 + the square root of two.

2. -2/3 sin 3pi/2 + 3/5 sin 5pi/3 - 6/7 sin 7pi/6
Here I keep on getting 2/3... I converted the expression to 2/3 + 0 - 0. However, my book says the answer is something completely different: 230- 63 . square root of 3/210

I feel completely clueless on how to do these... Did I not understand anything in math class?
Any help would be greatly appreciated!

 

[stapel]

1. 3 sin pi/2 - 2 sin 5pi/4 + 1/2 sin pi

For this one I begun by converting a few sines: 3- 2 sin 5pi/4 + 0. Am I doing something completely wrong? And if what I did so far is right, how do I calculate 2 sin 5pi/4?

You're right so far. Now use what you know about the sine of 5pi/4, sine on pi/4, and the shape and sign of the sine wave over the basic period.

2. -2/3 sin 3pi/2 + 3/5 sin 5pi/3 - 6/7 sin 7pi/6

Here I keep on getting 2/3... I converted the expression to 2/3 + 0 - 0. However, my book says the answer is something completely different: 230- 63 . square root of 3/210

How are you getting that the sine of (5/3)pi (which is the same, up to sign, as the sine of pi/3) as being zero?

 

[Emerald123]

Is the sine of pi/4= 1/2? Now I'm getting -2 for expression 1 :?
I don't understand exactly what you mean by "shape and sign of the sign wave over the basic period"... I'm sorry if this complicates things a lot! This is probably because I go to a school in Brazil, (though I am American), and everything is taught in Portuguese.

For expression 2 I multiplied 3/5 by 5pi/3 and got pi (I did the same for 6/7 sin 7pi/6); and then since the sine of pi= 0, I said that they were both equal to zero.

 

[stapel]

Is the sine of pi/4= 1/2?

No:

. . . . .\(\displaystyle \sin\left(\dfrac{\pi}{4}\right)\, =\, \dfrac{1}{\sqrt{\strut 2\,}}\)

This is one of the basic reference-angle values that you'll need to memorize.

I don't understand exactly what you mean by "shape and sign of the sign wave over the basic period"

Sorry; they were supposed to have taught you the basics of the sine and cosine ratios (here), along with their signs (here) and their graphs, both on the unit circle (here) and "horizontally" (here). So you'll need to learn about that stuff first, including memorizing the basic reference-angle values (here).

For expression 2 I multiplied 3/5 by 5pi/3 and got pi

The "sin" in the middle is not a variable or a factor; it is the name of the function. The 5pi/3 inside the function cannot be multiplied by the 3/5 outside the function. That's not how functions work! :shock:

 

[Emerald123]

Oh, pi/4= 45 degrees; I don't know why I didn't think of that before!
I tried doing both of them again, but I'm still getting wrong answers...
My resolution to #1:

And for #2:

I drew the circles to help me visualize the sines and to see whether they were positive or negative... Are they correct?
I read the links and they were very helpful! My teacher had never shown us an example of a sine wave graph. There are some things that I still don't understand well though: Why does the radius always remain positive in the example circles? And why is it that the circle is divided in two positive and two negative quadrants for the sine, cosine, etc.?... How is it that the side of a triangle can be negative in the circles?