Geometry B 2

Find the coordinates of the point of intersection of the angle 300º and the unit circle.

A

Tomas, Ling, and Daniel are experimenting with a giant rubber band. They each hold the rubber band to create a triangle. The distance from Tomas to Ling is 24 inches. The distance from Ling to Daniel is 36 inches. The distance from Daniel to Tomas is 20 inches. Find the measures of the three angles in the triangle.

A

Which vector has a direction of 49° north of west?

A

Write an equation for a circle with a diameter that has endpoints at (-1, -6) and (-3, -2). Round to the nearest tenth if necessary.

A

Write an equation for a circle with center at (-9, 5) and diameter = 24.

A

Zack, Rachel, and Maddie are unraveling a huge ball of yarn to see how long it is. As they move away from each other, they form a triangle. The distance from Zack to Rachel is 3 meters. The distance from Rachel to Maddie is 2.5 meters. The distance from Maddie to Zack is 4 meters. Find the measures of the three angles in the triangle.

A

unit circle

A circle that has a radius of 1 and is centered at the origin of the coordinate plane is a unit circle.

angle of rotation

A figure formed by a rotating ray and a stationary reference ray.

vector

A mathematical quantity that has magnitude, a numerical measure, and direction.

secant

A secant is any line that intersects a circle in exactly two points.

vector

A vector is a quantity, drawn as an arrow, with both direction and magnitude. For example, force and velocity are vectors. If a quantity has magnitude but not direction, it is called a scalar. Temperature, length, and mass are examples of scalars.

intercepted arc

An angle intercepts an arc if and only if each of the following conditions are met: the endpoints of the arc lie on an angle and each side of the angle contains an endpoint of the arc.

Use the law of cosines and the law of sines to solve for all missing parts of triangle ABC when side a = 20, side b = 12, and side c = 14.

B

Find the coordinates of P', the image of point P(1, 0) rotated 210° about the origin.

C

Find the coordinates of the point of intersection of the angle -225º and the unit circle.

C

In the figure, which relationship is not true? The diagram is not drawn to scale.

C

Which diagram shows how to find the sum of using the head-to-tail method?

C

Write the equation of the circle with the given center and a point on the circle. Then graph the circle. center: (3, 2) a point on the circle: (5, 2)

C

If the angle between each blade of the wind turbine is the same, and each blade has a length of 20 meters, find the distance between the tips of two blades. Round the answer to the nearest hundredth, if necessary.

C. 34.64 meters

Find the coordinates of the point of intersection of the angle -315º and the unit circle.

D

Use the parallelogram method to find the vector sum . Use the law of cosines to find , the magnitude of . Use the law of sines to find the angle that makes with.

D

Which vector has a direction of 32° west of north?

D

cosine

In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse represents the cosine.

acute case

The acute case states that the law of cosines is true for an acute triangle.

direction of a vector

The direction of a vector is the directed angle between the positive x-axis and the vector.

magnitude of a vector

The magnitude of a vector is the length of a vector (a numerical measure).

Write an equation for a circle with center at (-6, 10) and diameter = 6.

a. (x + 6)^2 + (y - 10)^2 = 9

Quadrilateral WRTZ is inscribed in a circle. If m<W = 45 and m<R = 100, find m<T m<Z.

a. 135 and 80

Find x. Assume that any segment that appears to be tangent is tangent.

a. 15

Find x. Assume that any segment that appears to be tangent is tangent.

a. 22

A swimmer heads in a downstream direction at an angle of 15° with the direction of a 2 mph current. The speed of the swimmer in still water is 2.5 mph. Find the swimmer's actual speed.

a. 4.46 mph

Find all angles between 0° and 180° for which sin = 0.866.

a. 60° and 120°

Find the measure of the numbered angle.

a. 70°

Give two values of between 0° and 180° for the given value of sin. Express your answers to the nearest degree. 0.9990

a. 87° and 93°

Describe the magnitude and direction of the vector or vectors in the following situation: a boat going 15 knots upstream against a current of 3 knots

a. The magnitude of the boat is 15 knots and the direction is upstream. The magnitude of the current is 3 knots and the direction is downstream.

, given the following measures, find the measure of the missing side.

a. b 6.7

Two inscribed angles of a circle that intercept the same arc are __________.

a. congruent

The magnitude of a vector is represented by its __________.

a. length

What is the value of cos 120°?

b. -0.5

What is the value of sin 330°?

b. -0.5

Use a calculator to find cos 295º. Round to four decimal places.

b. 0.4226

Use a calculator to find sin 65º. Round to four decimal places.

b. 0.9063

Dan is investigating a boat sunk at the mouth of a river. To reach the wreck, Dan must swim against the current of 2.7 mph. Suppose Dan dives and starts swimming downward at 4.1 mph (his speed in still water) at a 15° angle with the water's surface. Find the actual speed, s, Dan will swim as a result of the current.

b. 1.65 mph

For the given pair of vectors and , use the parallelogram method to find the vector sum . Use the law of cosines to find , the magnitude of . Use the law of sines to find the angle that makes with .

b. 10.71; 22.39º

In D, AB = CB and m arc CE = 54. Find m<BCE

b. 108

Trapezoid ABCD is inscribed in a circle. If m<A = 60, M<C = 120, and m<D = 60, find the m<B.

b. 120

Find the measure of the numbered angle.

b. 125°

Find an angle T on the unit circle such that 90° < T < 360°, tan T = tan U, and m<U=34º.

b. 214°

Use the diagram below to solve the problem. Note: The diagram is not drawn to scale.

b. 25°

If sin = 0.4756, what are all possible values of between 0° and 360°, rounded to the nearest degree?

b. 28° and 152°

Find x. Assume that any segment that appears to be tangent is tangent.

b. 40

In which quadrants is sin negative?

b. III and IV

Solve ΔABC with A = 61°, b = 33, and c = 39.

b. a = 36.91, B = 51.45°, and C = 67.55°

A plane is flying at 168 miles per hour with a heading of 43.5° from due north. The wind is blowing a constant 33 miles per hour at 133.5° from due north. Find the ground speed and true course of the plane.

b. ground speed: 171.2 mph; course: 54.6° from due north

Determine whether the statement is always, sometimes, or never true. A secant-tangent angle passes through the center of the circle.

b. sometimes

Determine whether the statement is always, sometimes, or never true. A vertex of a secant-tangent angle is a point on the circle.

b. sometimes

An airplane is flying due east at 180 km/hr while the wind is pushing it due north at 33 km/hr. What is the plane's resultant speed?

c. 183 km/hr

Give two values of between 0° and 180° for the given value of sin. Express your answers to the nearest degree. 0.7071

c. 45° and 135°

Find the value of y.

c. 97

In which quadrants is cos positive?

c. I and IV

Solve ΔABC with A = 65°, b = 36, and c = 32.

c. a = 36.69, B = 62.78°, C = 52.22°

A ship is moving in the water at approximately 20 mph. Vector AC represents the ship's speed. The current of the sea is going at a speed of approximately 15 mph. The sea's current is represented by vector AD. What is the resultant speed of the ship, vector AB?

c. approximately 25 mph

Find the unknown measures in the triangle.

c. mA 69.6°, mB 64.3°, mC 46.1°

Determine whether the statement is always, sometimes, or never true. A tangent to a circle passes through the center of the circle.

c. never

What is the value of tan 90°?

c. undefined

Apollo 8 was the first manned spacecraft to orbit the Moon at an average altitude of 185 km above the Moon's surface. Determine an equation to model a single circular orbit of the Apollo 8 command module if the radius of the Moon is 1,740 km. Let the center of the Moon be at the origin.

d

Write an equation for a circle with center at (6, -2) and diameter = 12.

d. (x-6)^2 + (y+2)^2= 36

Find the unknown measures in the triangle.

d. mA 34°

Find x.

d. 100

Find the measure of the numbered angle formed by a secant segment and tangent.

d. 110°

Find x, the measure of the angle formed. Assume that any segment that appears to be tangent is tangent.

d. 15

Find x. Assume that any segment that appears to be tangent is tangent.

d. 30

Find x. Assume that any segment that appears to be tangent is tangent.

d. 43

A boat leaves port and sails in still water at 5.6 mph for 3.5 hours. The boat then changes direction and sails at 4.9 mph for 4.25 hours. Then the boat turns and heads directly to port at 6 mph, taking 3.22 hours to reach port. At what angle will the boat have to turn to head directly back to the port?

d. 58.3°

Members of the soccer team are trying to map out some new plays before their next game. The goal is 24 feet wide. Sophie's favorite play has a player standing 18 feet from one goal post and 21 feet from the other post. In the triangle formed by the posts and the player, what is the measure of the angle formed opposite the goal?

d. 75.5°

given the lengths of the sides, find the measure of the stated angle to the nearest degree.

d. 78

Give two values of between 0° and 180° for the given value of sin. Express your answers to the nearest degree. 0.9900

d. 82° and 98°

, given the lengths of the sides, find the measure of the stated angle to the nearest degree.

d. 87

Solve Δ ABC with A = 70°, b = 30, and c = 51

d. a = 49.54, B = 34.68°, C = 75.32°

Find the unknown measures in the triangle.

d. mA 42.5°, mB 80°, mC 57.5°