Find each value if cos A=12/13 with 0 (degrees) <= A <= 90 (degrees) and if sin B=8/17 with 90 (degrees) <= B <=180 (degrees)
1. sin(A+B)
2. cos(A-B)
3. tan(A-B)
draw the two triangles in standard position.
A is a 5-12-13 triangle, and B is 8-15-17.
sinA = 5/13
cosA = 12/13
tanA = 5/12
sinB = 8/17
cosB = -15/17
tanB = -8/15
Now just plug those values into your addition formulas for the trig functions.
1. Well, to find sin(A+B), we first need to know what A and B are. But since we don't have the exact values, we'll just use their ratios. We know that sin B = 8/17, so we can call B the angle whose opposite side is 8 and whose hypotenuse is 17. As for A, we have cos A = 12/13, making A the angle whose adjacent side is 12 and whose hypotenuse is 13.
Now, we can use the angle addition formula for sine, which is sin(A+B) = sin A * cos B + cos A * sin B. We can plug in the ratios we have:
sin(A+B) = (12/13 * cos B) + (cos A * 8/17)
But wait, we also need the value of cos B. You didn't provide that. I can't do my math without all my numbers, just like a clown without her red nose. So, please provide the value of cos B, and we'll get back to the math party!
2. Oh boy, cos(A-B) is next up on our humor-filled journey. To find this value, we need to know the values of cos A and sin B, which you've kindly provided. Given cos A = 12/13, we know that A is the angle whose adjacent side is 12 and whose hypotenuse is 13. And, since sin B = 8/17, we can call B the angle whose opposite side is 8 and whose hypotenuse is 17.
Now for the fun part! We'll use the angle subtraction formula for cosine, which states that cos(A-B) = cos A * cos B + sin A * sin B. Let's plug in our value sandwiches:
cos(A-B) = (12/13 * cos B) + (cos A * 8/17)
But hold on tight, because we need the value of cos B to complete this hilarious calculation. Please provide that, and we'll continue our math circus!
3. Get ready for some good times with tangent! To find tan(A-B), we need to know the values of cos A and sin B, which you've graciously given us. Let's dive into this amusement park of mathematics!
Given cos A = 12/13, we know that A is the angle whose adjacent side is 12 and whose hypotenuse is 13. And since sin B = 8/17, we can call B the angle whose opposite side is 8 and whose hypotenuse is 17.
Now, let's use the angle subtraction formula for tangent, which tells us that tan(A-B) = (tan A - tan B) / (1 + tan A * tan B). So let's bring out the clowns and calculate:
tan(A-B) = (tan A - tan B) / (1 + tan A * tan B)
But wait, my funny friend! We need the values of both tan A and tan B to proceed with this math-filled comedy show. You only provided cos A and sin B, leaving us with half a joke. So, if you'd be so kind, provide the tangents, and we'll continue our hilarious numerical journey!
To find the solutions for sin(A+B), cos(A-B), and tan(A-B), we will need to use the trigonometric identities. Firstly, let's find the missing trigonometric values for A and B.
Given that cos A = 12/13, we can use the Pythagorean identity sin^2 A + cos^2 A = 1 to find sin A. Plugging in the given value, we have:
sin^2 A + (12/13)^2 = 1
sin^2 A = 1 - (12/13)^2
sin^2 A = 1 - 144/169
sin^2 A = 169/169 - 144/169
sin^2 A = 25/169
sin A = √(25/169) (Taking the positive square root since 0 ≤ A ≤ 90)
sin A = 5/13
Given that sin B = 8/17, we can again use the Pythagorean identity to find cos B:
sin^2 B + cos^2 B = 1
(8/17)^2 + cos^2 B = 1
64/289 + cos^2 B = 1
cos^2 B = 1 - 64/289
cos^2 B = 289/289 - 64/289
cos^2 B = 225/289
cos B = √(225/289) (Taking the positive square root since 90 ≤ B ≤ 180)
cos B = 15/17
Now, let's calculate the values for the given expressions:
1. sin(A+B):
Using the sum of angles identity, sin(A+B) = sin A * cos B + cos A * sin B
sin(A+B) = (5/13) * (15/17) + (12/13) * (8/17)
sin(A+B) = 75/221 + 96/221
sin(A+B) = 171/221
2. cos(A-B):
Using the difference of angles identity, cos(A-B) = cos A * cos B + sin A * sin B
cos(A-B) = (12/13) * (15/17) + (5/13) * (8/17)
cos(A-B) = 180/221 + 40/221
cos(A-B) = 220/221
3. tan(A-B):
Using the difference of angles identity, tan(A-B) = (tan A - tan B)/(1 + tan A * tan B)
tan(A-B) = (5/12 - 8/15) / (1 + (5/12) * (8/15))
tan(A-B) = (25/60 - 32/60) / (1 + 40/180)
tan(A-B) = (-7/60) / (1 + 2/9)
tan(A-B) = (-7/60) / (11/9)
tan(A-B) = -63/660
tan(A-B) = -21/220
Therefore, the values are:
1. sin(A+B) = 171/221
2. cos(A-B) = 220/221
3. tan(A-B) = -21/220
To find the values of the trigonometric functions in the given ranges, we need to use the given information about cosine and sine values and apply trigonometric identities to find the required values.
1. To find sin(A + B), we can use the trigonometric identity:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Given that cos(A) = 12/13 and sin(B) = 8/17, we can substitute these values into the identity:
sin(A + B) = (sin(A) * (12/13)) + ((12/13) * (8/17))
Now, we need to find sin(A):
We can use the Pythagorean identity: sin^2(A) + cos^2(A) = 1
Rearranging the equation, we get: sin^2(A) = 1 - cos^2(A)
Substituting cos(A) = 12/13 into the equation, we get:
sin^2(A) = 1 - (12/13)^2
Taking the square root of both sides, we get:
sin(A) = ±sqrt(1 - (12/13)^2)
Since 0 (degrees) <= A <= 90 (degrees), we know that A is in the first quadrant where sin(A) is positive. Therefore:
sin(A) = sqrt(1 - (12/13)^2)
Plug in the values:
sin(A + B) = (sqrt(1 - (12/13)^2) * (12/13)) + ((12/13) * (8/17))
2. To find cos(A - B), we can use the trigonometric identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Given that cos(A) = 12/13 and sin(B) = 8/17, we can substitute these values into the identity:
cos(A - B) = (cos(A) * (12/13)) + ((sqrt(1 - (12/13)^2)) * (8/17))
Now, we need to find sqrt(1 - (12/13)^2) using the same method as in step 1.
3. To find tan(A - B), we can use the trigonometric identity:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
So, we need to find tan(A) and tan(B):
tan(A) = sin(A) / cos(A)
tan(B) = sin(B) / cos(B)
Substitute the given values into the equations:
tan(A) = sqrt(1 - (12/13)^2) / (12/13)
tan(B) = (8/17) / sqrt(1 - (8/17)^2)
Plug in these values into the tan(A - B) equation:
tan(A - B) = (sqrt(1 - (12/13)^2) / (12/13) - (8/17) / sqrt(1 - (8/17)^2)) /
(1 + (sqrt(1 - (12/13)^2) / (12/13)) * (8/17) / sqrt(1 - (8/17)^2))