I do not know which value I am solving for or how to get there these are the instructions and info given.
Use the given conditions to find the exact value of the expression.
sinx=(-5/13), tanx>0, sin(x-(pi/3))
There is no question and the answer is ((12(sqrt3))-5)/26 from the back of the book. Help please!
To find the exact value of the expression sin(x - π/3), we can use the following steps:
Step 1: Calculate the value of cos(π/3) using the unit circle or trigonometric identities. We know that cos(π/3) = 1/2.
Step 2: Use the trigonometric identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
Step 3: Substitute the values sin(x) = -5/13, tan(x) > 0, and cos(π/3) = 1/2 into the identity sin(x - π/3) = sin(x)cos(π/3) - cos(x)sin(π/3).
Substituting the values into the expression:
sin(x - π/3) = (-5/13)(1/2) - cos(x)(√3/2)
Step 4: We need to find the value of cos(x). Since tan(x) > 0 and sin(x) = -5/13, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos(x).
Squaring both sides of the equation sin(x) = -5/13 and using the identity, we get:
(-5/13)^2 + cos^2(x) = 1
25/169 + cos^2(x) = 1
cos^2(x) = 1 - 25/169
cos^2(x) = 144/169
cos(x) = ± √(144/169) = ± 12/13
However, we know that tan(x) > 0, so cos(x) must be positive.
Thus, cos(x) = 12/13.
Step 5: Substitute the value of cos(x) back into the expression:
sin(x - π/3) = (-5/13)(1/2) - (12/13)(√3/2)
= -5/26 - 12√3/26
= (-5 - 12√3)/26
Therefore, the exact value of sin(x - π/3) is (-5 - 12√3)/26 or ((12√3) - 5)/26 (same value, just rearranged).
To solve this problem, let's break it down step by step:
1. We are given the following conditions: sin(x) = -5/13, tan(x) > 0, and sin(x-(pi/3)).
2. We need to find the exact value of the expression sin(x-(pi/3)).
3. Recall the trigonometric identity for the sine of the difference of two angles: sin(a - b) = sin(a)cos(b) - cos(a)sin(b).
4. In this case, we are given sin(x - (pi/3)), so we need to determine the values of sin(x) and cos(x).
5. From the first condition, sin(x) = -5/13.
6. To find cos(x), we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
7. Plugging in the given value of sin(x), we have (-5/13)^2 + cos^2(x) = 1.
8. Simplifying, 25/169 + cos^2(x) = 1. Subtracting 25/169 from both sides, we get cos^2(x) = 144/169.
9. Taking the square root of both sides, we have cos(x) = 12/13 (since cos(x) > 0 because tan(x) > 0 is given).
10. Now that we know sin(x) = -5/13 and cos(x) = 12/13, we can substitute these values into the sine difference identity: sin(x - (pi/3)) = sin(x)cos(pi/3) - cos(x)sin(pi/3).
11. Since cos(pi/3) = 1/2 and sin(pi/3) = √3/2, we have sin(x - (pi/3)) = (-5/13)(1/2) - (12/13)(√3/2).
12. Simplifying, we get sin(x - (pi/3)) = -5/26 - 12√3/26.
13. Combining the terms, we have sin(x - (pi/3)) = (-5 - 12√3)/26.
14. This matches the expression you provided from the back of the book: ((12√3) - 5)/26.
Therefore, the exact value of the expression sin(x - (pi/3)) is ((12√3) - 5)/26.