find the exact value of sin 2a, given that sin a + 5/13, with the angle a in quadrant II?

Question

HELP

Answer 1

Do you mean sina = 5/13?

If you do then sin 2a = 2sin a . cos a

sin² a + cos² a = 1
cos² a = 1 - sin² a
= 1 - 25/169
= 144/169

So cos a = ±12/13

Since a is a 2nd quadrant angle cos a <0
Thus cos a = -12/13

Thus sin 2a = 2 . 5/13 . -12/13
= -120/169

Answer 2

Given Sin a = 5/13 and a is in quadrant 2 tells that Cos a = -12/13

Note Quad 2 Sin is +ve and Cos and Tan are -ve. Recall that 5 12 13 form a Pythagorean triple and so form a RIGHT ANGLED TRIANGLE.

Now

Sin 2a = 2 Sina Cos a
= 2 5/13 -12/13
= -120/169

Answer 3

I assume you mean "sin a = 5/13".

a is in quadrant II, so pi-a is in quadrant I. Let a'=pi-a. Since sin(a') = sin(a), it follows that a' is the angle of a right triangle, with the ratio of the side opposite a', which I'll call O, to the hypotenuse, H, being 5:13. We then assume that the lengths of these sides are O=5 and H=13.

By the Pythagorean Theorem, the length of the side A adjacent to angle a' satisfies O^2 + A^2 = H^2. Plug in O=5 and H=13, and you get
5^2 + A^2 = 13^2
=> 25 + A^2 = 169
=> A^2 = 144
so A = 12.

By SOHCAHTOA, cos(a') = A/H = 12/13.

sin(2a') = 2*sin a'*cos a'. Therefore, sin(2a') = 2*(5/13)*(12/13) = 120/169.

Finally, sin(2a) = sin(2(pi-a')) = sin(2pi-2a') = -sin(2a') = -120/169.

This comes from the fact that sin(2pi+x) = sin(x), and sin(-x) = -sin(x).

Answer 4

Do mean sina 5/13
sin 2a = 2sin a . cos a

sin² a + cos² a = 1
cos² a = 1 - sin² a
= 1 - 25/169
= 144/169
cos a = ±12/13
2nd quadrant angle cos a <0
cos a = -12/13
sin 2a = 2 . 5/13 . -12/13
= -120/169

Answer 5

I'm assuming that you meant sin a = 5/13.

Okay, here's the easy way...

sin a = 5/13
Take the arcsin to determine a...
a = acrsin(5/13) = 22.62 degrees.

However, the problem says that a is in quadrant II. A as solved above is in quadrant I. Therefore, we need to find the angle in quadrant II that corresponds to 22.62 degrees.

a = 180 degrees - 22.62 degrees = 157.38 degrees

Now, we can determine what the sin of 2a is...

sin (2a) = sin (2*157.38) = -0.71

Answer 6

sin2a=2sina cosa
given sina=5/13
so cosa =12/13
sin2a=2(5/13)(12/13)
=120/169
it is ih the first quadrant