Find the exact value of the trigonometric function given that
sin u = 5/13 and cos = -3/5
1) sin(u-v)
My teacher said to make two right triangles and then find the missing parts by the pyth theorem.
For the 5/13 triangle I got 12 and the other triangle I got 4. I do not know what to do now.
so the angle u could be in quadrant I inside a 5:12:13 triangle, with 13 as the hypotenuse,
so cos u = 12/13
angle v could be in quadrants II or III and are inside the 3:4:5 triangle , with 5 the hypotenuse, so
sin v = 4/5
draw the triangle in quadrant II with those dimensions
You must by now have learned that
sin(A-B) = sinAcosB - cosAsinB
then sin(u-v) = sinucosv - cosusinv
= (5/13)(-3/5) - (12/13)(4/5) = -63/65
Thank You
To find the value of sin(u-v), we can use the trigonometric identity:
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
Since we are given sin(u) and cos(v), we need to find cos(u) and sin(v) to substitute into the formula.
From the first triangle where sin(u) = 5/13, we can use the Pythagorean theorem to find the missing side length:
a^2 + b^2 = c^2
a^2 + (5/13)^2 = 1^2
a^2 + 25/169 = 1
a^2 = 144/169
a = 12/13
From the second triangle where cos(v) = -3/5, we can also use the Pythagorean theorem to find the missing side length:
a^2 + b^2 = c^2
(3/5)^2 + b^2 = 1^2
9/25 + b^2 = 1
b^2 = 16/25
b = 4/5
Now we have all the necessary values to calculate sin(u-v):
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
= (5/13)(-3/5) - (12/13)(4/5)
= -15/65 - 48/65
= -63/65
Therefore, the exact value of sin(u-v) is -63/65.
To find the value of sin(u-v), where u is the given angle and v is another unknown angle, you can use the trigonometric identity:
sin(u-v) = sin u * cos v - cos u * sin v
Based on the given information: sin u = 5/13 and cos u = -3/5, we need to find the value of cos v and sin v.
To do that, you can construct two right triangles to represent sin u = 5/13 and cos u = -3/5.
For the first triangle with sin u = 5/13:
- The opposite side (the side opposite to angle u) is 5, and
- The hypotenuse (the longest side) is 13.
For the second triangle with cos u = -3/5:
- The adjacent side (the side adjacent to angle u) is -3, and
- The hypotenuse is 5.
Now, using the Pythagorean theorem, we can find the remaining sides of each triangle. Let's solve for the first triangle:
a^2 + 5^2 = 13^2
a^2 + 25 = 169
a^2 = 169 - 25
a^2 = 144
a = 12
For the second triangle:
b^2 + (-3)^2 = 5^2
b^2 + 9 = 25
b^2 = 25 - 9
b^2 = 16
b = 4
Now that we have the remaining sides of each triangle, we can find cos v and sin v.
cos v = adjacent/hypotenuse
sin v = opposite/hypotenuse
For the first triangle:
cos v = -3/5 (already given)
sin v = 12/13
Now, let's substitute these values into the trigonometric identity sin(u-v) = sin u * cos v - cos u * sin v:
sin(u-v) = (5/13) * (-3/5) - (-3/5) * (12/13)
sin(u-v) = -15/65 + 36/65
sin(u-v) = 21/65
Therefore, the exact value of sin(u-v) is 21/65.