5sin + 12cos = 13 , find the value of tan
making table
Angle, left side
0 ,,,,, 12
30 ,,,, 12.89
40 ,,,, 12.41
hmm
24 ,,,, 12.996 <---- around 24 degrees
25 ,,,, 12.99
26 ,,,, 12.97
27 ,,,, 12.96
28 ,,,, 12.94
29 ,,,, 12.91
If 5 sin+12cos=13 find the value of tan
To find the value of tan, we can use the trigonometric identity:
tan(x) = sin(x) / cos(x)
From the equation given, 5sin + 12cos = 13, we can rearrange it to solve for sin:
5sin = 13 - 12cos
sin = (13 - 12cos) / 5
Now, we need to find cos in order to calculate tan. We can use the identity:
sin^2(x) + cos^2(x) = 1
Substituting the value of sin from above:
[(13 - 12cos) / 5]^2 + cos^2 = 1
Simplifying the equation:
(169 - 312cos + 144cos^2) / 25 + cos^2 = 1
Multiplying through by 25 to eliminate the denominator:
169 - 312cos + 144cos^2 + 25cos^2 = 25
Combining like terms:
169 - 312cos + 169cos^2 = 25
Rearranging the equation:
169cos^2 - 312cos + 144 = 0
Now we have a quadratic equation in terms of cos. We can solve this equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
cos = (-b ± sqrt(b^2 - 4ac)) / 2a
Where a = 169, b = -312, and c = 144.
cos = (-(-312) ± sqrt((-312)^2 - 4(169)(144))) / (2 * 169)
cos = (312 ± sqrt(97200 - 92592)) / 338
cos = (312 ± sqrt(4624)) / 338
cos = (312 ± 68) / 338
cos = 380 / 338 or 244 / 338
cos = 1.1243 or 0.7221
Now, we can substitute these values of cos back into the equation sin = (13 - 12cos) / 5 to find sin:
For cos = 1.1243:
sin = (13 - 12(1.1243)) / 5
sin ≈ -0.5542
For cos = 0.7221:
sin = (13 - 12(0.7221)) / 5
sin ≈ 0.5739
Finally, we can calculate tan using the values of sin and cos:
For sin ≈ -0.5542:
tan = sin / cos ≈ -0.5542 / 1.1243 ≈ -0.4932
For sin ≈ 0.5739:
tan = sin / cos ≈ 0.5739 / 0.7221 ≈ 0.7944
Therefore, the two possible values of tan are approximately -0.4932 and 0.7944.
To find the value of tan, we can use the Pythagorean Identity, which states that sin^2(x) + cos^2(x) = 1. From the given equation: 5sin + 12cos = 13, we can rearrange it to isolate sin:
5sin = 13 - 12cos
Dividing both sides by 5, we have:
sin = (13 - 12cos)/5
Now, substitute this value of sin into the Pythagorean Identity:
((13 - 12cos)/5)^2 + cos^2 = 1
Expand and simplify the equation:
(169 - 312cos + 144cos^2)/25 + cos^2 = 1
Multiply through by 25 to eliminate the denominator:
169 - 312cos + 144cos^2 + 25cos^2 = 25
Combine like terms:
169 + 169cos^2 - 312cos = 25
Rearrange the equation:
169cos^2 - 312cos + 144 = 0
Now, we can solve this quadratic equation for cos using the quadratic formula:
cos = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 169, b = -312, and c = 144, so we can substitute the values into the quadratic formula:
cos = (-(-312) ± √((-312)^2 - 4(169)(144))) / 2(169)
Simplifying:
cos = (312 ± √(97200 - 97344)) / 338
cos = (312 ± √(-144)) / 338
Here, we have a negative value under the square root, which means there are no real solutions for cos. Therefore, there is no real value for tan in this equation.