Domanda del libro di testo
If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 38
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. C = 102°, a = 16 meters, b = 20 meters
Guida verificata passo dopo passo1
Identify the given elements of the triangle: angle \(C = 102^\circ\), side \(a = 16\) meters, and side \(b = 20\) meters. We need to find the area of the triangle.
Recall the formula for the area of a triangle when two sides and the included angle are known: \(\text{Area} = \frac{1}{2}ab \sin C\).
Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 16 \times 20 \times \sin 102^\circ\).
Calculate \(\sin 102^\circ\) using a calculator or trigonometric tables to find the sine of the given angle.
Multiply the values together to find the area, then round the result to the nearest whole number as requested.
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Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful for finding an unknown side or angle when two sides and the included angle are known, or vice versa. In this problem, it helps confirm the triangle's dimensions or find missing elements if needed.
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Percorso guidato
4:35
Intro to Law of Cosines
Area of a Triangle Using Two Sides and Included Angle
The area of a triangle can be calculated using the formula (1/2)ab sin(C), where a and b are two sides and C is the included angle. This formula is especially useful when the height is unknown but two sides and the included angle are given, as in this problem.
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Percorso guidato
4:02Calculating Area of SAS Triangles
Trigonometric Functions and Angle Measurement
Understanding how to use trigonometric functions like sine and cosine with angles measured in degrees is essential. Correctly applying sin(102°) ensures accurate calculation of the area. Familiarity with converting and using angles in degrees is important for precise results.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 2j, w = 3i + 6j
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Domanda del libro di testo
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
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Domanda del libro di testo
In Exercises 37–39, find the dot product v ⋅ w. Then find the angle between v and w to the nearest tenth of a degree.
v = 2i + 4j, w = 6i - 11j
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Domanda del libro di testo
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||w - u||
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Domanda del libro di testo
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i
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