Pre Calculus

Question

Use the given triangle to evaluate the trigonometric expression. The graph is a right triangle, the side on the y-axis =5 and the side o the x-axis =12. The hypotenuse is not given. sin(y/2)=?

a) 2/√3

b) 4/13

c) 6/13

d) √8/13

Answer 1

There is another possibility, but there could be a misprint in the answer.

 

Let a = 5, be the length on the y- axis

Let b = 12, be the length on the x-axis

 

Again, the hypotenuse would be c = 13, by the Pythagorean Theorem.

 

Suppose y is the angle that is made between the y-axis and the hypotenuse.

 

From this, cos(y) = 5/13.

 

From the half-angle formula, sin(y/2) = √((1 - cos(y))/2)

 

With the given value of cos(y), sin(y/2) = 2/√13

This would answer a, if it was misprinted.

 

Without seeing the actually drawing, there might be a misprint in the question or the answers given above.

                                     

Answer 2

You have a right triangle with leg lengths of 5 and 12.
The hypotenuse will be 13.
This is from the Pythagorean Theorem:  a²+b²=c²
5₂+12²=c²
169 = c²
√169 = c
13 = c
 
However; since there is no picture here... I'm not sure
what your "y/2" signifies in sin(y/2) = ?

*******************************************

 

We can find the angles that are missing from the triangle.

We have a 90 degree angle ... The other 2 angles must

sum to 90 degrees.

 

Taking the horizontal line of length 12...

And using the hypotenuse length of 13

cos x = 12/13

Take inverse cosine of each side

 

x = cos^-1(12/13) ≅ 22.62 degrees

 

The other angle will be 180 - (90+22.62) ≅ 67.38 degrees

 

That is the angle created between the vertical line up to (0,5)

and the hypotenuse running diagonally from there to (12,0)

 

If your problem is misprinted and is actually

 

[sin(y)]/2

sin(67.38)/2≅ .461538

 

and

 

6/13 ≅ .461538

 

I believe the answer you are looking for is c ... 6/13