Trigonometry Help: Evaluating sin2y & cos2y

[huntingrdr]

Homework Statement

If sin x = 1/3 and sec y = 5/4, where x and y lie between 0 and pi/2, evaluate the expression.

sin2y

cos2y

Homework Equations

2siny = 2siny*cosy?

cos2y = cos^2 y - sin^2 y?

The Attempt at a Solution

I think sin2y goes to 2siny*cosy. I know what sin x is but how do I find sin y? The answer should be 24/25 for siny, but not sure how to get it. Not sure what the answer is or how to get it for cos2y.

 

[Mark44]

First thing you should do is to draw two right triangles, one with an angle x and the other with an angle y. In the first triangle, you have that sin x = 1/3, so make the side opposite angle x 1 unit, and make the hypotenuse 3 units. You should be able to figure out the length of the side adjacent to angle x, so you can determine cos x.

For the other triangle, you're given that sec y = 5/4, so cos y = 4/5 (sec y = 1/(cos y)). In that triangle, label the hypotenuse 5 units and the adjacent side 4 units. What does the side opposite y need to be? Once you find it, you can find sin y.

 

[Architeuthis Dux]

Thank you for reaching out for assistance with your trigonometry homework. To evaluate sin2y and cos2y, we first need to use the given information to find the values of sin y and cos y.

Using the identity sin^2 y + cos^2 y = 1, we can solve for sin y as follows:

sin^2 y + cos^2 y = 1

(1/3)^2 + cos^2 y = 1

1/9 + cos^2 y = 1

cos^2 y = 1 - 1/9

cos^2 y = 8/9

Taking the square root of both sides, we get:

cos y = √(8/9)

cos y = 2√2/3

Similarly, using the identity sec y = 1/cos y, we can solve for cos y as follows:

sec y = 5/4

1/cos y = 5/4

cos y = 4/5

Now, we can plug these values into the expressions for sin2y and cos2y as follows:

sin2y = 2sin y * cos y

= 2 * (1/3) * (2√2/3)

= 4√2/9

cos2y = cos^2 y - sin^2 y

= (4/5)^2 - (1/3)^2

= 16/25 - 1/9

= 135/225

= 9/15

= 3/5

Therefore, the final answers for sin2y and cos2y are 4√2/9 and 3/5, respectively. I hope this helps you understand the process for evaluating these expressions. Good luck with your homework!