Chapter 7

Represent the complex number graphically, and find the trigonometric form of the number. $$-7+4 i$$

Find the area of the triangle having the indicated angle and sides. \(C=110^{\circ}, \quad a=6, \quad b=10\)

Use vectors to find the interior angles of the triangle with the given vertices. $$(-3,-4), (1,7), (8,2)$$

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$a=11, \quad b=13, \quad c=7$$

Represent the complex number graphically, and find the trigonometric form of the number. $$5-3 i$$

Use vectors to find the interior angles of the triangle with the given vertices. $$(-3,0), (2,2), (0,6)$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector. $$\mathbf{u}=\langle 4,2\rangle, \mathbf{v}=\langle 8,0\rangle$$

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$A=42^{\circ}, \quad B=35^{\circ}, \quad c=1.2$$

Find the area of the triangle having the indicated angle and sides. \(A=150^{\circ}, \quad b=8, \quad c=10\)

Use vectors to find the interior angles of the triangle with the given vertices. $$(-3,5), (-1,9), (7,9)$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=\langle 5,3\rangle, \mathbf{v}=\langle-4,0\rangle$$

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$C=95^{\circ}, \quad b=19, \quad c=25$$

Represent the complex number graphically, and find the trigonometric form of the number. $$6$$

Find the area of the triangle having the indicated angle and sides. \(C=120^{\circ}, \quad a=4, \quad b=6\)

Find \(\mathbf{u} \cdot \mathbf{v},\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) $$\|\mathbf{u}\|=9,\|\mathbf{v}\|=36, \theta=\frac{3 \pi}{4}$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=\langle-6,-8\rangle, \mathbf{v}=\langle 2,4\rangle$$

Use Heron's Area Formula to find the area of the triangle. $$a=12, \quad b=24, \quad c=18$$

Represent the complex number graphically, and find the trigonometric form of the number. $$3+\sqrt{3} i$$

Find the area of the triangle having the indicated angle and sides. \(B=75^{\circ} 15^{\prime}, \quad a=103, \quad c=58\)

Find \(\mathbf{u} \cdot \mathbf{v},\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) $$\|\mathbf{u}\|=4,\|\mathbf{v}\|=12, \theta=\frac{\pi}{3}$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=\langle 0,-5\rangle, \mathbf{v}=\langle-4,10\rangle$$

Use Heron's Area Formula to find the area of the triangle. $$a=25, \quad b=35, \quad c=32$$

Represent the complex number graphically, and find the trigonometric form of the number. $$-2 \sqrt{2}+i$$

Find the area of the triangle having the indicated angle and sides. \(C=85^{\circ} 45^{\prime}, \quad a=16, \quad b=20\)

Find \(\mathbf{u} \cdot \mathbf{v},\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) $$\|\mathbf{u}\|=4,\|\mathbf{v}\|=10, \theta=\frac{2 \pi}{3}$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=2 \mathbf{i}-3 \mathbf{j}$$

Use Heron's Area Formula to find the area of the triangle. $$a=5, \quad b=8, \quad c=10$$

Represent the complex number graphically, and find the trigonometric form of the number. $$-1-5 i$$

Find \(\mathbf{u} \cdot \mathbf{v},\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) $$\|\mathbf{u}\|=100,\|\mathbf{v}\|=250, \theta=\frac{\pi}{6}$$

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \mathbf{v}=-\mathbf{i}+\mathbf{j}$$

Use Heron's Area Formula to find the area of the triangle. $$a=12, \quad b=17, \quad c=8$$

Represent the complex number graphically, and find the trigonometric form of the number. $$1+3 i$$

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=\langle 10,-6\rangle\\\ &\mathbf{v}=\langle 9,15\rangle \end{aligned}$$

Use Heron's Area Formula to find the area of the triangle. $$a=1.24, \quad b=2.45, \quad c=1.25$$

Represent the complex number graphically, and find the trigonometric form of the number. $$5-2 i$$

A plane flies 500 kilometers with a bearing of \(316^{\circ}\) (clockwise from north) from Naples to Elgin. The plane then flies 720 kilometers from Elgin to Canton (see figure). Canton is due west of Naples. Find the bearing of the flight from Elgin to Canton.

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=\langle 12,4\rangle\\\ &\mathbf{v}=\left\langle\frac{1}{4},-\frac{1}{3}\right\rangle \end{aligned}$$

Use Heron's Area Formula to find the area of the triangle. $$a=2.4, \quad b=2.75, \quad c=2.25$$

Represent the complex number graphically, and find the trigonometric form of the number. $$-3+i$$

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of \(12^{\circ}\) with the horizontal. The flagpole casts a 16 -meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is \(20^{\circ} .\) (a) Draw a triangle that represents the problem. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation involving the unknown quantity. (c) Find the height of the flagpole.

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=\mathbf{j}\\\ &\mathbf{v}=\mathbf{i}-\mathbf{j} \end{aligned}$$

Use Heron's Area Formula to find the area of the triangle. $$a=1, \quad b=\frac{1}{2}, \quad c=\frac{3}{4}$$

Represent the complex number graphically, and find the trigonometric form of the number. $$3 \sqrt{2}-7 i$$

The angles of elevation \(\theta\) and \(\phi\) to an airplane are being continuously monitored at two observation points \(A\) and \(B,\) respectively, which are 2 miles apart, and the airplane is east of both points in the same vertical plane. (a) Draw a diagram that illustrates the problem. (b) Write an equation giving the distance \(d\) between the plane and point \(B\) in terms of \(\theta\) and \(\phi\) (c) Use the equation from part (b) to find the distance between the plane and point \(B\) when \(\theta=40^{\circ}\) and \(\phi=60^{\circ}\)

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}\\\ &\mathbf{v}=-\mathbf{i}-\mathbf{j} \end{aligned}$$

Use Heron's Area Formula to find the area of the triangle. $$a=\frac{3}{5}, \quad b=\frac{5}{8}, \quad c=\frac{3}{8}$$

Represent the complex number graphically, and find the trigonometric form of the number. $$-8-5 \sqrt{3} i$$

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=\langle 10,20\rangle\\\ &\mathbf{v}=\langle-18,9\rangle \end{aligned}$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{u}=\langle 6,0\rangle$$.

Represent the complex number graphically, and find the standard form of the number. $$6\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$