Question 1170014: Suppose sin x =2/7,cos x > 0, sin y = − 2/5 and cos y < 0. Then
cos x=
cos y=
Find each of the following quantities:
sin(x+y)=
Cos(x+y)=
Tan(x+y)=
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**1. Find cos x:**
* We know sin²x + cos²x = 1.
* sin x = 2/7, so sin²x = (2/7)² = 4/49.
* cos²x = 1 - sin²x = 1 - 4/49 = 45/49.
* cos x = ±√(45/49) = ±(3√5)/7.
* Since cos x > 0, cos x = (3√5)/7.
**2. Find cos y:**
* We know sin²y + cos²y = 1.
* sin y = -2/5, so sin²y = (-2/5)² = 4/25.
* cos²y = 1 - sin²y = 1 - 4/25 = 21/25.
* cos y = ±√(21/25) = ±√21/5.
* Since cos y < 0, cos y = -√21/5.
**3. Find sin(x+y):**
* sin(x+y) = sin x cos y + cos x sin y
* sin(x+y) = (2/7)(-√21/5) + ((3√5)/7)(-2/5)
* sin(x+y) = (-2√21)/35 - (6√5)/35
* sin(x+y) = (-2√21 - 6√5)/35
**4. Find cos(x+y):**
* cos(x+y) = cos x cos y - sin x sin y
* cos(x+y) = ((3√5)/7)(-√21/5) - (2/7)(-2/5)
* cos(x+y) = (-3√(105))/35 + 4/35
* cos(x+y) = (4 - 3√105)/35
**5. Find tan(x+y):**
* tan(x+y) = sin(x+y) / cos(x+y)
* tan(x+y) = ((-2√21 - 6√5)/35) / ((4 - 3√105)/35)
* tan(x+y) = (-2√21 - 6√5) / (4 - 3√105)
**Final Answers:**
* cos x = (3√5)/7
* cos y = -√21/5
* sin(x+y) = (-2√21 - 6√5)/35
* cos(x+y) = (4 - 3√105)/35
* tan(x+y) = (-2√21 - 6√5) / (4 - 3√105)
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