I cannot figure these two problems out. Thank you for you help, I really appreciate it!
5. Solve the triangle (see the image.)
6. Calculate the exact value of sin(x + y) if sin(x) = 1/3 and angle x ends in the second quadrant and cos(y) = 1/5 and angle y is in the first quadrant.
5. Since the angles `alpha ` and `beta ` are given, the third angle of the triangle can be found using the fact that the sum of angles in a triangle is 18 degrees:
`alpha + beta + gamma = 180 `
`41+ 33 + gamma=180 `
`gamma =180-33-41=106 `
Now, since the side c is given, we can use the Sine theorem to find remaining sides. The Sine theorem says that in a triangle all ratios of sides to the sines of opposite angles are equal:
`a/sin(alpha) = b/sin(beta) = c/sin(gamma) `
`c/sin(gamma) = 21.85 `
From here, we can solve for a and b:
`a=21.85*sin(alpha) =14.33 `
`b=21.85*sin(beta) =11.9 `
So, the unknown quantities in the given triangle are
`gamma=106 ` degrees
a = 14.33, b = 11.9
6. We need to use the Sine of Sum formula to solve this problem:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
We already know sin(x) and cos(y), but we don't know cos(x) and sin(y). They can be found using Pythagorean theorem:
`sin^2(x) + cos^2(x) = 1 `
`cos(x) = +-sqrt(1-sin^2(x)) = +-sqrt(1-1/9) = +-sqrt(8)/3=+-(2sqrt(2))/3 `
Since angle x is in the second quadrant, its cosine is negative, so
` cos(x) = -(2sqrt2)/3`
Similarly, for angle y
`sin^2(y) + cos^2(y) = 1 `
`sin(y) = +-sqrt(1-cos^2(y)) = +-sqrt(1-1/25)) = +-sqrt(24)/5 = +-(2sqrt6)/5 `
Since angle y is in the first quadrant, its sine is positive.
`sin(y) = (2sqrt(6))/5 `
Plug in these values into the formula:
`sin(x + y) = 1/3 * 1/5 + (-2sqrt(2))/3 * (2sqrt(6))/5 = 1/15 - (4sqrt(12))/15=(1- 8sqrt(3))/15 `
The exact answer for sin(x + y) is ` (1-8sqrt(3))/15` .
6. x-ends in the second quadrant. sine function is positive here, and cosine - negative. y-ends in the first quadrant, where both sine and cosine functions are positive. Again, (x+y) must end in the third quadrant where sine function is negative.
Applying the trigonometric identity `sin^2theta+cos^2theta=1 `
From the formula of sum of angles,
As expected, this function has a negative value.