Given that sin(a) = 2/3 and cos(b) = −1/5, with a and b both in the interval [𝜋/2, 𝜋), find the exact values of sin(a + b) and cos(a − b).

Question

Given that sin(a) = 2/3 and cos(b) = −1/5, with a and b both in the interval [𝜋/2, 𝜋), find the exact values of sin(a + b) and cos(a − b).

Answer 1

Make sketches of two triangles with the given data, both must be

in quadrant II

sin a = 2/3 = opposite/hypotenuse
you triangle is right-angled with y = 2 and r = 3
x^2 + y^2 = r^2
x^2 + 4 = 9
x = ±√5, but from the given domain, we know x = -√5
so cos a = -√5/3

for the 2nd triangle, cos b = -1/5, so x, the adjacent = -1, hypotenuse = 5
(-1)^2 + y^2 = 25
y = √24 and sin b = √24/5 or 2√6/5

then sin(a+b) = (sina)(cosb) + cosa(sinb)
=(2/3)(-1/5) + (-√5/3)(2√6/5)
= -2/15 - 2√30/15
= -(2 + 2√30)/15

from you expansion of cos(a - b)
follow the same steps, you already know the sines and cosines of a and b

Answer 2

Well, well, well, looks like we have some trigonometry fun ahead! Let's tackle this one step at a time.

First, let's find sin(a + b). To do that, we need to use the sum formula for sine:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

We already know the values of sin(a) and cos(b), so we can substitute them in:
sin(a + b) = (2/3)(-1/5) + cos(a)sin(b)

Now, we need to find cos(a) and sin(b) to complete the equation. Since a and b are both in the interval [𝜋/2, 𝜋), we know that cos(a) and sin(b) are negative. Let's find the values:

cos(a) = - √(1 - sin²(a))
cos(a) = - √(1 - (2/3)²)
cos(a) = - √(1 - 4/9)
cos(a) = - √(5/9)
cos(a) = - √5/3

sin(b) = √(1 - cos²(b))
sin(b) = √(1 - (-1/5)²)
sin(b) = √(1 - 1/25)
sin(b) = √(24/25)
sin(b) = √24/5

Now, let's substitute these values back into the equation:
sin(a + b) = (2/3)(-1/5) + (-√5/3)(√24/5)

Whoa-ho-ho, math is really turning into a circus here! Let's simplify this big top expression:

sin(a + b) = -2/15 - √(120/75)
sin(a + b) = -2/15 - √(8/5)
sin(a + b) = -2/15 - 2√(5)/5

Phew! That's quite a show for sin(a + b). Now, let's move on to cos(a - b).

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

We already know the values of cos(a) and sin(b), so let's substitute those in:
cos(a - b) = (-√5/3)(-1/5) + (2/3)(√24/5)

Hold onto your big floppy shoes, we're about to simplify another expression!

cos(a - b) = √5/15 + 2√(24)/15
cos(a - b) = √5/15 + 2√(4*6)/15
cos(a - b) = √5/15 + 2√(4)*√(6)/15
cos(a - b) = √5/15 + 2*2*√(6)/15
cos(a - b) = √5/15 + 4√(6)/15
cos(a - b) = (√5 + 4√(6))/15

And there we have it, my friend! The exact values of sin(a + b) and cos(a - b) are:

sin(a + b) = -2/15 - 2√(5)/5

cos(a - b) = (√5 + 4√(6))/15

Hope you enjoyed the circus of math tricks!

Answer 3

To find the exact values of sin(a + b) and cos(a - b), we can use the trigonometric sum and difference formulas.

1. First, let's find the value of sin(a + b):
sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

2. Substitute the given values:
sin(a + b) = (2/3) * (-1/5) + cos(a) * sin(b)

3. To find cos(a), we can use the identity sin^2(a) + cos^2(a) = 1:
sin^2(a) + cos^2(a) = 1
(2/3)^2 + cos^2(a) = 1
4/9 + cos^2(a) = 1
cos^2(a) = 1 - 4/9
cos^2(a) = 5/9
cos(a) = ± √(5/9)

4. Since a is in the interval [𝜋/2, 𝜋), we know that cos(a) is negative:
cos(a) = - √(5/9)

5. Now let's find sin(b):
cos^2(b) + sin^2(b) = 1
(-1/5)^2 + sin^2(b) = 1
1/25 + sin^2(b) = 1
sin^2(b) = 1 - 1/25
sin^2(b) = 24/25
sin(b) = ± √(24/25)

6. Since b is in the interval [𝜋/2, 𝜋), we know that sin(b) is positive:
sin(b) = √(24/25)

7. Substitute the values of cos(a) and sin(b) into our equation for sin(a + b):
sin(a + b) = (2/3) * (-1/5) + (- √(5/9)) * (√(24/25))
sin(a + b) = -2/15 - (√(5/9)) * (√(24/25))
sin(a + b) = -2/15 - (√(5) * √(24))/(√(9) * √(25))
sin(a + b) = -2/15 - (√(5) * √(24))/(3 * 5)
sin(a + b) = -2/15 - (√(5) * 2√(6))/(3 * 5)
sin(a + b) = -2/15 - (2√(5∙6))/(3 * 5)
sin(a + b) = -2/15 - (2√(30))/(3 * 5)
sin(a + b) = -2/15 - (√30)/15
sin(a + b) = (-2 - √30)/15

8. Therefore, the exact value of sin(a + b) is (-2 - √30)/15.

9. To find the value of cos(a - b), we can use the trigonometric difference formula:
cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

10. Substitute the given values:
cos(a - b) = (- √(5/9)) * (-1/5) + (2/3) * (√(24/25))
cos(a - b) = √(5/9) * (1/5) + (2/3) * (√(24/25))
cos(a - b) = √(5)/√(9) * 1/√(5) + 2/√(3) * √(24)/√(25)
cos(a - b) = 1/√(9) + 2 * √(24)/3√(25)
cos(a - b) = 1/3 + 2 * √(24)/(3 * 5)
cos(a - b) = 1/3 + √(24)/15
cos(a - b) = (1 + √24)/15

11. Therefore, the exact value of cos(a - b) is (1 + √24)/15.

Answer 4

To find the value of sin(a + b), we can use the sum formula for sine:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

Given that sin(a) = 2/3 and cos(b) = -1/5, we can substitute these values into the formula:
sin(a + b) = (2/3)(-1/5) + cos(a) * sin(b)

To find the value of cos(a - b), we can use the difference formula for cosine:
cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

Let's calculate each of these:

For sin(a + b):
sin(a + b) = (2/3)(-1/5) + cos(a) * sin(b)
= -2/15 + cos(a) * sin(b)

Now we need to find the values of cos(a) and sin(b).

Since a is in the interval [𝜋/2, 𝜋), sin(a) is positive and cos(a) is negative. Since sin(a) = 2/3, we can use the Pythagorean identity to solve for cos(a):
cos(a) = sqrt(1 - sin^2(a))
= sqrt(1 - (2/3)^2)
= sqrt(1 - 4/9)
= sqrt(5/9)
= sqrt(5) / 3

Now that we have cos(a), we can substitute it back into the equation for sin(a + b):
sin(a + b) = -2/15 + (sqrt(5)/3) * sin(b)

For sin(b), we need to find the value in the interval [𝜋/2, 𝜋). Since cos(b) = -1/5, we can use the Pythagorean identity to solve for sin(b):
sin(b) = sqrt(1 - cos^2(b))
= sqrt(1 - (-1/5)^2)
= sqrt(1 - 1/25)
= sqrt(24/25)
= sqrt(24) / 5

Now we can substitute the value of sin(b) back into the equation for sin(a + b):
sin(a + b) = -2/15 + (sqrt(5)/3) * (sqrt(24) / 5)
= -2/15 + (sqrt(5) * sqrt(24)) / (3 * 5)
= -2/15 + (sqrt(5 * 24) / 15)
= -2/15 + (sqrt(120) / 15)
= (-2 + sqrt(120)) / 15
= (-2 + 2 * sqrt(30)) / 15

Therefore, the exact value of sin(a + b) is (-2 + 2 * sqrt(30)) / 15.

Now let's find the value of cos(a - b):
cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

We already know the values of cos(a), sin(b), and cos(b), so we can substitute these into the equation:
cos(a - b) = (sqrt(5)/3) * (-1/5) + (2/3) * (sqrt(24)/5)
= -sqrt(5) / 15 + (2 * sqrt(6)) / 15
= (-sqrt(5) + 2 * sqrt(6)) / 15

Therefore, the exact value of cos(a - b) is (-sqrt(5) + 2 * sqrt(6)) / 15.