Exam 7: Applications of Trigonometry

The trigonometric form of complex numbers is often used by electrical engineers to describe the current I, voltage V, and impedance Z in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115-volt, alternating current. The relationship among these three quantities is $I = \frac { V } { Z }$ . The real part of I represents the actual current delivered to an electrical appliance in amps. Approximate this current when $V = 164 \operatorname { cis } 42 ^ { \circ }$ and $Z = 102 \operatorname { cis } 11 ^ { \circ }$ , and express the answer in rectangular form to two decimal places.

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $5 i$ and $4 j$

Represent the complex number geometrically. $5$

Use De Moivre's theorem to change the given complex number to the form $a + b i$ , where $a$ and $b$ are real numbers. $( - 2 - 2 t ) ^ { 10 }$

Determine whether the given pair of vectors are orthogonal or not. $\{ 4,1 \}$ and $\{ 1 , - 4 \}$

A helicopter hovers at an altitude that is $d = 979$ feet above a mountain peak of altitude 5,184 feet, as shown in the figure. A second, taller peak is viewed from both the mountaintop and the helicopter. From the helicopter, the angle of depression is $43 ^ { \circ }$ , and from the mountaintop, the angle of elevation is $18 ^ { \circ }$ . Approximate the altitude of the taller peak.

Use De Moivre's theorem to change the given complex number to the form $a + b i$ , where $a$ and $b$ are real numbers. $( - 1 + i ) ^ { 24 }$

Two tugboats are towing a large ship into port, as shown in the figure. The larger tug exerts a force of 4,100 pounds on its cable, and the smaller tug exerts a force of 3,500 pounds on its cable. If the ship is to travel on a straight line l, approximate the angle $0$ that the larger tug must make with l.

Given that $a = \langle 3 , - 2 \rangle , b = \langle 3,3 \rangle , \text { and } c = \langle - 2,2 \rangle$ , find the number. $a \cdot ( b + c )$

If c represents a constant force, find the work done if the point of application of c moves along the line segment from P to Q. $c = 6 i + 6 i$ and $P ( 0,0 ) , Q ( 5 , - 3 )$

A rhombus has sides of length 100 centimeters, and the angle at one of the vertices is 70 $0$ . Approximate the lengths of the diagonals to the nearest tenth of a centimeter.

Solve $\triangle A B C .$ $\beta = 121.624 ^ { \circ } , b = 0.218 , c = 0.149$

Shown in the figure is a plan for the top of a wing of a jet fighter, where $a = 22.3$ feet and $b = 18.9$ feet. Approximate angle $\varphi$ .

Given that $a = \langle 4 , - 1 \rangle , b = \langle 2,3 \rangle , \text { and } c = \langle - 5,3 \rangle$ , find the number. $( a - b ) \cdot ( b + c )$

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $\{ - 10,4 \}$ and $( 8,4 )$

The volume V of the right triangular prism shown in the figure is $\frac { 1 } { 3 } B h$ where B is the area of the base and h is the height of the prism. Approximate h, if $d = 11.3$ .

Solve $\triangle A B C .$ $y = 70 ^ { \circ } , c = 11 , b = 12$

Find 4a + 6b. a = < 5, - 4>, b = <6, 5 >

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $( - 4,3 )$ and $\{ - 16,12 )$

Use De Moivre's theorem to change the given complex number to the form $a + b i$ , where $a$ and $b$ are real numbers. $( 5 + 5 i ) ^ { 5 }$