Theta is an acute angle and sin theta= 3/7 and cos theta = -2 square root 10/7?

Question

I've done the math out for it to come out to be
3(-2 square root 10/ -2 square root 10

I know I messed up with the rationalizing and it's still not complete...I'm forgetting the rules with square roots....can anyone help??

Answer 1

For one thing, if theta is acute, its cosine is positive. You can draw a right triangle with theta as one of the acute angles, opposite side 3, hypotenuse 7 and calculate the adjacent leg using the Pyhtagorean theorem. It is sqrt 40 = 2sqrt10. So cos theta = (2 sqrt 10) / 7. I think you are looking for tan theta (???). With this triangle you drew, you can make it really easy on yourself and just use opposite over adjacent as the tangent ratio = 3/(2sqrt10) = (3sqrt10) / 20.

Since you said you had trouble rationalizing, you just multiply 3 / (2sqrt10) by sqrt10/sqrt10.

Answer 2

This is what you'd do if you wanted to find cos(theta) from sin(theta) without using a calculator. Now I'm not sure you actually want that... anyway, this method might still be helpful to you at some point

Draw a triangle

Label an acute angle theta
Sin(theta) = opposite over hypotenuse, so opp side is length
3 and hypotenuse is length 7.

Use pythagoras to get the other side to be sqrt(7^2 - 3^2)
= sqrt(40) = 2 * sqrt(10)

cos(theta) = adjacent over hypotenuse
= 2* sqrt(10) / 7

Edit: listen to the guy after me lol

Answer 3

Impossible! cos θ is always positive if θ is acute.
Let's do it right.
sin θ = 3/7 so sin² θ = 9/49.
cos² θ = 1 - 9/49 = 40/49
Taking the square root(which must be positive),
cos θ = √40/7 = (2/7) *√10.
So your original answer is right, except for sign.

Answer 4

What is it that you're supposed to solve for? tan(theta)?

your cosine term is incorrectly expressed, it should be -2*sqrt(10)/7, since sin^2 + cos^2 = 1

Tan(theta) is sin/cos = -3/(2sqrt(10)) , multiply top and bottom by sqrt(10) = -3sqrt(10)/20

Answer 5

is the square root include (10/7)? or is it (sq. rt. 10)/7. and are u allowed to use calculator?

Answer 6

4.38