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Chapter 5: Problem 16

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$B=75^{\circ} 20^{\prime}, \quad a=6.2, \quad c=9.5$$

Short Answer

Expert verified
After carrying out these calculations, we will have the measures of all angles and lengths of sides in the triangle. The values obtained would be specific to the provided measurements, but the steps used can be generalized for any problem of this nature.

Step by step solution

01

Convert angle to decimal degrees

First, we need to convert angle B from degrees and minutes to decimal degrees. The conversion is done using the formula: Degrees + minutes/60. So, \(B = 75 + \frac{20}{60} = 75.33^{\circ}\)
02

Apply law of cosines to find side b

Now, we can apply the law of cosines to find side b. The Law of cosines is \(b^2 = a^2 + c^2 - 2ac\cos(B)\). Plugging in the known values gives \(b^2 = (6.2)^2 + (9.5)^2 - 2*(6.2)*(9.5) *\cos(75.33)\). Solving this equation gives us the length of side b.
03

Use law of cosines to find angle A

After finding side b, we can use the law of cosines again to find angle A. The law of cosines to find the angle when all sides of the triangle are known is \( \cos(A) = \frac{b^2+c^2-a^2}{2bc}\). Substituting the known values will yield the cosine of angle A. Taking the inverse of the cosine gives us angle A in radians, and we need to convert this to degrees by multiplying by \(\frac{180}{\pi}\).
04

Use the property of triangles to find angle C

The sum of all angles in a triangle is always \(180^{\circ}\) . So we can find angle C by subtracting angle A and angle B from \(180^{\circ}\). This gives us the measurement of angle C.

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Most popular questions from this chapter

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sec x+\tan x-x$$ Trigonometric Equation $$\sec x \tan x+\sec ^{2} x-1=0$$

Fill in the blank. \(\cos (u+v)=\)_____

Consider the equation \(2 \sin x-1=0\). Explain the similarities and differences between finding all solutions in the interval \(\left[0, \frac{\pi}{2}\right)\), finding all solutions in the interval \([0,2 \pi),\) and finding the general solution.

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x}=4$$

The table shows the average daily high temperatures in Houston \(H\) (in degrees Fahrenheit) for month \(t,\) with \(t=1\) corresponding to January. (Source: National Climatic Data Center) $$ \begin{array}{|c|c|} \hline \text { Month, } t & \text { Houston, } \boldsymbol{H} \\ \hline 1 & 62.3 \\ 2 & 66.5 \\ 3 & 73.3 \\ 4 & 79.1 \\ 5 & 85.5 \\ 6 & 90.7 \\ 7 & 93.6 \\ 8 & 93.5 \\ 9 & 89.3 \\ 10 & 82.0 \\ 11 & 72.0 \\ 12 & 64.6 \\ \hline \end{array} $$ (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above \(86^{\circ} \mathrm{F}\) and below \(86^{\circ} \mathrm{F}\).
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