Ch. 4 - Laws of Sines and Cosines; Vectors
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- In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.
Problem 1
- Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. cos(45° - 30°)
Problem 1
- Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- < α < 𝝅 5 2 5 𝝅 cos β = ------ , 0 < β < ------ 13 2 Find the exact value of each of the following. cos (α + β)
Problem 1
- Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. sin 6x sin 2x
Problem 1
- In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 3i + j, w = i + 3j
Problem 1
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.
Problem 1
- In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
Problem 1
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
Problem 2
- In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.
Problem 2
- Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. cos 7x cos 3x
Problem 3
- In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j
Problem 3
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.
Problem 3
- In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.
Problem 3
- In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12
Problem 3
- In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
Problem 3
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
Problem 4
- If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.
Problem 4
Problem 4.14
Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (2, -5), P₂ = (-6, 6)
Problem 4.18
Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (-1, 6), P₂ = (7, -5)
Problem 4.33
The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.
Problem 4.40
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = -5j
Problem 4.42
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 8i - 6j
Problem 4.44
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 4i - 2j
Problem 4.46
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = i - j
- In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w
Problem 5
- In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. cos 50° cos 20° + sin 50° sin 20°
Problem 5
- In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. cos 50° cos 20° + sin 50° sin 20°
Problem 5
- In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = 3i + j
Problem 5
- In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
Problem 5
- In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w
Problem 6