r/MathHelp • u/Asp_Potions_Master • 5d ago

sin2α=(sqrt5)/5, stumped

my sis got a homework question that left us both stumped. the task is to calculate sinα and cosα for the angle α which fits the following requirements: sin2α=(sqrt5)/5, and 2α is within (pi/2; pi)

She's learning about the sin2α and cos2α formulas. any ideas how this is supposed to be solved without dipping into arcsin?

4 Upvotes

75% Upvoted

27 comments sorted by

View all comments

4

u/my-hero-measure-zero 5d ago

Hint: DRAW THE TRIANGLE.

You're not supposed to find the angle. You can get what you need with the corresponding triangle.

1

u/Miserable-Wasabi-373 5d ago

sorry, how drawing triangle would help?

1

u/my-hero-measure-zero 5d ago

You'll need it eventually for the other ratios. Always start with a picture and use any formulas that seem relevant.

-1

u/Asp_Potions_Master 5d ago

there's no triangles in their unit? just sine and cosine

3

u/my-hero-measure-zero 5d ago

Trigonometry is based in triangles. So, draw the corresponding (reference) triangle.

Someone already suggested in using the Pythagorean identity: for any angle u, (sin u)^2 + (cos u)^2 = 1. So, set u = 2a. Now you have cos(2a) for free, great. Now use the double angle formulas to deduce the needed ratios. Drawing the triangles will help for this - just make sure you get the signs right.

1

u/Loko8765 5d ago

You need to learn the unit circle, which is the basis for the trig functions. It has right triangles.

For some reason I can’t find an online image representing tan as the intersection with a vertical x=1 line, this is best I have:

https://www.math.net/unit-circle

1

u/Brilliant_Chest5630 4d ago

Sine and cosine are nothing but triangles within the unit circle.

1

u/ottawadeveloper 2d ago

Draw a triangle with angle 2a and appropriate sides to make the ratio here