SOLUTION: Use the given conditions to find the exact value of the expression. sin α = − 5/13, tan α > 0, sin (α − 5π/3) I got 5+{{{sqrt(3)}}}/3 which is wrong. This is my w

Algebra ->  Trigonometry-basics -> SOLUTION: Use the given conditions to find the exact value of the expression. sin α = − 5/13, tan α > 0, sin (α − 5π/3) I got 5+{{{sqrt(3)}}}/3 which is wrong. This is my w      Log On


   

Question 1159666: Use the given conditions to find the exact value of the expression.
sin α = − 5/13, tan α > 0, sin (α − 5π/3)
I got 5+sqrt%283%29/3 which is wrong.
This is my work:
cos a=-sqrt%281-sin%5E2+%28theta%29%29= 12/13
b= -5pi/3 which is has the values (-1/2) for cosb and (sqrt%283%29/2 for sinb.
I use sin(a-b) formula= sinAcosB+cosAsinB
= (-5/13)(-1/2)+(12/13)(sqrt%283%29/2)= 5+12sqrt%283%29/36 = 5+sqrt%283%29/3 or am I not suppose to simplify the 12 and 36?

Found 4 solutions by solver91311, ikleyn, MathLover1, math_helper:
Answer by solver91311(24713) About Me  (Show Source):
Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.

In this site, there is a group of lessons, where you can find solutions to many similar problems
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
    - Evaluating trigonometric expressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Trigonometry: Solved problems".


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Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


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There is nothing EXTRAORDINARY with such problems . . .

You simply should know the technique of their solutions, and you can learn this technique from the referred lessons.



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Use the given conditions to find the exact value of the expression.
sin+%28alpha%29+=+-5%2F13
tan+%28alpha%29+%3E+0+
sin+%28alpha+-+5pi%2F3%29

cos%28alpha%29=-sqrt%281-sin%5E2%28alpha%29+%29
cos%28alpha%29=-sqrt%281-%28-5%2F13%29%5E2+%29
cos%28alpha%29=-sqrt%281-25%2F169+%29
cos%28alpha%29=-sqrt%28169%2F169-25%2F169+%29
cos%28alpha%29=-sqrt%28144%2F169+%29
cos%28alpha%29=-12%2F13

tan+%28alpha%29+=+%28-5%2F13+%29%2F%28-12%2F13%29+
+tan+%28alpha%29+=+5%2F12=>tan+%28alpha%29+%3E+0+


sin+%28alpha+-+5pi%2F3%29->use

sin+%28alpha+-+beta%29=sin%28alpha%29+cos%28beta%29+-+cos%28alpha%29sin%28beta%29+

let beta=5pi%2F3, already given sin+%28alpha%29+=+-5%2F13+, and cos%28alpha%29=-12%2F13

sin+%28alpha+-+5pi%2F3%29=-%285%2F13%29cos%285pi%2F3%29+-+%28-12%2F13%29+sin%285pi%2F3%29....cos%285pi%2F3%29=1%2F2, sin%285pi%2F3%29=-sqrt%283%29%2F2


sin+%28alpha+-+5pi%2F3%29=-5%2F26+-%286sqrt%283%29%29%2F13
sin+%28alpha+-+5pi%2F3%29=-5%2F26+-%286sqrt%283%29%29%2F13




Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!


In sin(a-b) = sin(a)cos(b)-cos(a)sin(b), the minus sign is provided, so

b is not -5pi/3;  instead,  it is simply 5pi/3  [ it might help to think of it as a pattern match. You are subtracting one angle from another, but both those angles have their own signs. ]



cos(b) is +1%2F2+

sin(b) is -sqrt%283%29%2F2



resulting in 





Although you had the sign error, there was also an error in how you simplified the expression-- you somehow lost the 1/26 from the first term.  Note that sin() and cos() will always produce outputs between -1 and +1 inclusive.  The "5+" in your answer is an immediate indication that something is wrong. 



But, I want to congratulate you on your effort.  Not many students post their work here and it is refreshing to see... AND an "A" for your approach.  





EDIT: I fixed the sin(a-b) formula and two missing minus signs, result is the same.