Textbook Question
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
B = 125°, a = 8 yards, c = 5 yards
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 4, Problem 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
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
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
Textbook Question
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
Textbook Question
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
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||w - u||
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i