Question

If sin α = − 5/13 and tan α > 0, find the exact value of sin(α − 5π/3).

Answer 1

you gotta use the subtraction angle formula for sine
do you know it?

Answer 2

if the answer is "no i have no clue" that is fine, i will write it for you

Answer 3

btw this problem is not hard, but it is a pain in the ...

Answer 4

is it sin(a - b) = (sina)(cosb) -(cosa)(sinb) ??

Answer 5

yes

Answer 6

so what is missing is the \(\cos(\alpha)\)

Answer 7

all the rest you know

Answer 8

if the sin of 5pi/3 = sqrt-(3)/2, is the cos(a) = 1/2 ?

Answer 9

No I don't! Help me set the problem up??

Answer 10

\[\sin(\frac{5\pi}{3})=-\frac{\sqrt3}{2}\]
\[\cos(\frac{5\pi}{3})=\frac{1}{2}\]
\[\sin(\alpha)=-\frac{5}{13}\] you have 3 of the four numbers

Answer 11

the one that is missing is
\[\cos(\alpha)\] which you get by drawing a triangle

Answer 12

all that you need is the missing side
do you know what it is?

Answer 13

First you're told that sine is negative and tangent is positive. This happens in the third quadrant.

Now using Pythagorean theorem, find the third side length of this triangle.

Answer 14

The third side = 12?

Answer 15

Then use the idntity\[\sin (A -B)=\sin A \cos B -\cos A \sin B\]

Answer 16

Yes but it's -12 (the third side because it extends on the negative x-axis). Understand @lizbens8 ??

Answer 17

you can think of it as 12 or -12, but \[\cos(\alpha)=-\frac{12}{13}\]

Answer 18

now plug in the numbers is all that is left to do
you have all four numbers that you need

Answer 19

\[\sin(\alpha-\beta) = (5\pi/3)(12/13) - (1/2 - \sin \beta) ??\]

Answer 20

oh no

Answer 21

just put the numbers in the formula

Answer 22

\[\sin (\alpha -\beta)=\sin (\alpha)\cos (\beta) -\cos (\alpha) \sin (\beta)\]
you know all four numbers

Answer 23

NO. You mean\[\sin (\alpha - \frac{ 5\pi }{ 3})=\sin \alpha \cos \frac{ 5\pi }{ 3 }-\cos \alpha \sin \frac{ 5\pi }{ 3 }\]

Answer 24

\[\sin(\frac{5\pi}{3})=-\frac{\sqrt3}{2}\\
\sin(\alpha)=-\frac{5}{13}\\
\cos(\frac{5\pi}{3})=\frac{1}{2}\\
\cos(\alpha)=-\frac{12}{13}\] as you said earlier

Answer 25

put the numbers in the right spot is all

Answer 26

What doesn't make sense is that the angle 5pi/3 is in the fourth quadrant and in the question you wrote that tangent of alpha > 0. I think either you copied it incorrectly or the textbook had it wrong but either way it should be tangent < 0.

Answer 27

Okay so \[-5 - 12\sqrt{3}/26\] is what I got

Answer 28

i didn't do it but it looks reasonable
want me to check?

Answer 29

\[-\frac{5}{13}\times \frac{1}{2}-(-\frac{12}{13})\times (-\frac{\sqrt3}{2})\] is the first step

Answer 30

so yeah, what you got

Answer 31

I checked on my homework answers and that was it! Thank you!

Answer 32

yw, hope it was more or less clear