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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 8: Q. 38 (page 535)

In the given problem solve the triangle using either the law of sines or law of cosines- $a = 4, c = 5, B = 55 掳$

Short Answer

Expert verified

Required values of the triangle are $b = 4.25, A = 50.45 掳, C = 74.55 掳$

Step by step solution

01

Step 1. Given information

Given values are $a = 4, c = 5, B = 55 掳$ .

We know for a triangle with sides a, b, c and opposite angles A, B, C, respectively, the Law of Cosines are:

$c^{2} = a^{2} + b^{2} - 2 a b \cos C b^{2} = a^{2} + c^{2} - 2 a c \cos B a^{2} = b^{2} + c^{2} - 2 b c \cos A$

02

Step 2. Calculation

$F o r b,$

$b^{2} = a^{2} + c^{2} - 2 a c \cos B = 4^{2} + 5^{2} - 2 脳 4 脳 5 脳 \cos (55 掳) = 16 + 25 - 40 脳 0.5735 = 41 - 22.94 = 18.06 b = 4.25$

$F o r A,$

localid="1646578213238" $\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} = \frac{{(4.25)}^{2} + 5^{2} - 4^{2}}{2 脳 4.25 脳 5} = \frac{18.0625 + 25 - 16}{42.5} = 0.6367 A = \cos^{- 1} (0.6367) = 50.45 掳$

$F o r C,$

$C = 180 掳 - A - B = 180 掳 - 50.45 掳 - 55 掳 = 74.55 掳$

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Most popular questions from this chapter

A special case of the Law of Cosines is the Pythagorean theorem.

Constructing a Highway U.S. 41, a highway whose primary directions are north鈥搒outh, is being constructed along the west coast of Florida. Near Naples, a bay obstructs

the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay?

Find the real solutions, if any, of the equation $3 x^{2} + 1 = 4 x$ .

The displacement d (in meters) of an object at time t (in seconds) is given $d = - 3 \sin (\frac{1}{2} t)$

(a) Describe the motion of the object.

(b) What is the maximum displacement from its resting position?

(c) What is the time required for one oscillation?

(d) What is the frequency?

Mollweide鈥檚 Formula For any triangle, Mollweide鈥檚 Formula (named after Karl Mollweide, 1774鈥�1825) states that $\frac{a + b}{c} = \frac{\cos [\frac{1}{2} (A - B)]}{\sin (\frac{1}{2} C)}$

Derive it.

[Hint: Use the Law of Sines and then a Sum-to-Product Formula. Notice that this formula involves all six parts of a triangle. As a result, it is sometimes used to check the solution of a triangle.]

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