precalc h final 2022

Find the exact value: sin (2π/3 + 5π/6)

-1

csc (7π/6)

-2

Evaluate sin (11π/6)

mm

Given function value: tan (β) = 5 Find: tan (90° - β)

mm

Solve the system of equations: {-x+y=4 {x^2+y=3

mm

Use trigonometric identities to simplify: csc (θ) tan (θ)

mm

evaluate sin 31π/6

mm

Solve the system of linear equations. x-3y+z=1 2x-y-2z=2 x+2z-3z=-1

no solutions

Are the vectors u=<2,-3>u=<2,−3>and v=<6,4>v=<6,4> orthogonal?

orthogonal

Find all solutions of cos^4 (2x) = 9/16

±π/12 + K (π/2)

Write the complex number z = √8 [cos (-π/3) i sin (-π/3)] in standard form.

√2 - √6i

Find absolute value of z=-2+5i

√29

The dot product of u with itself is 5. What is the magnitude of u?

√5

Find the exact value of tan (arccos (2/3))

√5/2

Write an algebraic expression that is equivalent to the expression. cos (arcsin((x-h)/r))

√r² - (x-h)² / r

{x^2+4x-y=7x2+4x−y=7 {2x-y=-12x−y=−1

(-4,-7) and (2,5)

Perform the indicated operation and use the fundamental identities to simplify. (1/sec x+1) - (1/sec x-1)

-2cot²x

Solve the equation 3sinx+ 2=sinx for -π/2 ≤x≤ π/2

-π/2

find the value of arcsin (sin(5π/3))

-π/3

Solve equation 2 sin² x+ 3 sin x + 1 = 0

-π/6 + 2kπ, 7π/6 + 2kπ, 3π/2 + 2kπ

Find the reference angle θ' for θ=4.1

0.9584 radians

Verify the identity. tan²y (csc² y -1)

1

Find the one cube roots of z=-2+2i

1 + i

Find the exact value of cos 95° cos 35° + sin 95°sin 35°

1/2

Write cos 3x cos 2x as a sum or difference.

1/2 cos x + 1/2 cos 5x

Given the equation for simple harmonic motion t d=3 sin (t/2)

1/4π

Find magnitude of the vector v that has initial point (4,-7) and terminal point (-1,5)

13

A ladder leaning against a house reaches 24 feet up the side of the house. The ladder makes a 60° angle with the ground. How far is the base of the ladder from the house? Round your answer to two decimal places.

13.86 feet

Find an angle that is coterminal with θ = − π/8.

15π/8

find the product of z1z2 of the complex numbers: zi = 2 (cos (2π/3) + i sin (2π/3)) and z2 = 8 (cos (11π/6) + i sin (11π/6))

16i

Find the exact value of tan 15°.

2 - √3

Write cos 4x + cos 2x as a product.

2 cos 3x cos x

A photographer takes a picture of a three- foot painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting. The angle β is subtended by the camera lens x feet from the painting is β = arctan (3x/x² + 4) Approximate distance from the picture when β is maximum.

2 feet

Convert 9π/8 radians to degrees.

202.5°

Find the angle between u=<4,3> and v=<3,5>

22.2°

A force of 600 pounds is required to pull a boat and trailer up a ramp inclined at 15° from the horizontal. Find the combined weight of the boat and trailer.

2318 pounds

Convert 120° to radians

2π/3

Use the given values to evaluate sec(θ) (if possi-ble): cscθ = 2, tanθ = √3/3

2√3/3

Use the given values to evaluate tan(x)(if possi-ble) cos (π/2 - x) = 3/5 (x) = 4/5

3/4

Find the reference angle θ ′ for θ=210°

30°

Find the supplement of θ = π/4

3π/4

Use Demoivre's Theorem to find

4096

Use multiple-angle formulas to express cos 3x in terms of cos x.

4cos^3 x−3cosx

From 1998 to 2004, the population of Colorado increased more rapidly than the population of Alabama. Two models that approximate the populations P (in thousands) are p=3488+81.9t p=4248+19.9t where t represents the year, with t =8 corresponding to 1998. Use the two models to estimate the populations of both states in 2010.

5,126,000 and 4,646,000

A yacht is going 14 knots East for 3 hours, then turns N 42° E for an hour. How far from port is the yacht.

52.4

A satellite traveling in a circular orbit approximately 1800 km. above the surface of Earth takes 2.5 hrs. to make an orbit. The radius of the earth is approximately 6400 km. Approximate the linear speed of the satellite in kilometers per hour.

6560 π km / hr

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of ,where 0<θ<π/2 √64 - 16x² , x = 2 cos θ

8 sin θ

Find a unit vector in the direction of v=<-2,5>

< -2√29/29, 5√29/29>

Find the component form of the vector v that has initial point (4,-7) and terminal point (−1,5).

<-5, 12>

Let u=<-1,3>,v=<2,4>,w=<1,-2>. Find dot product (u∙v)w

<10, -20>

Let v=<-2,5>, and w=<3,4>. Find v+2w

<4,13>

Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle of 30° below the horizontal.

<−50√3​,−50>

Find the amplitude and the period of y=-4cos(3x)

A=4, P= 2π/3

Use the half-angle formulas to simplify the expression. √(1 + cos4x) / 2

I cos2x I

Derive a reduction formula for sin (t + (π/2)

cos (t)

Write the expression as the sine, cosine, or tangent of an angle. cos 60°cos 20° - sin 60°sin 20°

cos 80°

Simplify: sin(π/2 - x) / cos(π/2 - x)

cot(x)

Match the trigonometric expressions

cot(x)sin(x) = cos(x) sin(x)sec(x) = tan(x) sin(x)(csc(x)-sin(x)) = cos^2(x) csc(x)/cot(x) = sec(x)

Use trigonometric identities to simplify: tan(θ) + cot(θ) / tan(θ)

csc² (θ)

Perform the indicated operation and use the fundamental identities to simplify ((sinθ + cosθ) / sinθ) - ((cosθ - sinθ) / cosθ)

secθ cscθ

Angles of depression and elevation are equal.

true

The term angle of depression means . . . for objects that lie below the horizontal, the angle from the horizontal downward to an object.

true

The term angle of elevation means . . . the angle from the horizontal upward to an object.

true

Solve 2cos²x-1 = 0

x = π/4 + nπ, x = 3π/4 + nπ

Find a trigonometric function to model the data in the following table.

y=2sinx+2

Find the quotient z1/z2 of the complex numbers: z1 = 24 (cos 300° + i sin 300°) and z2 = 8 (cos 75° + i sin 75°)

(-3√2)/2 - (-3√2)/2 i

Solve the system of linear equations. x - 2y + z = 2 2x - y - z = 1

(a,a-1,a)

Let u be the vector with initial point (2,−5) and terminal point (−1,3). Write u as a linear combination of the standard unit vectors i and j.

-3i+8j

Find all solutions of cos(x - π/3) + cos(x + π/3) = 1 in the interval [0, 2π).

0

A small business invests $10,000 in equipment to produce a new soft drink. Each bottle of the soft drink costs $0.65 to produce and is sold for $1.20. How many bottles must be sold before the business breaks even?

18,182 bottles

A 15-inch diameter tire on a car makes 9.3 revolutions per second. Find the angular speed of the tire in radians per second.

18.6π radians per second.

Find dot product <4,5>∙<2,3>

23

To close a barn's sliding door, a person pulls on a rope with a constant force of 50 pounds at a constant angle of 60°. Find the work done in moving the door 12 feet to its closed position.

300 foot-pound

Use right triangles to evaluate the expression: sin (arccos (3/5) - arcsin (5/13))

33/65

Find the direction angle of vector v=3i-4j

53.13°

Find the five key points (intercepts, maximum points, and minimum points) of the graph of y=−4cos(3x)

(0, -4) , (π/6, 0)

Solve the system of linear equations. x-2y+3z=9 -x+3y+z=-2 2x-5y+5z=17

(1,-1,2)

Solve the system of linear equations. x-2y+3z=9 y+4z=7 z=2

(1,-1,2)

Solve the system of equations: {y=lnx {x+y=1

(1,0)

Verify the identity. √(1-cosx)(1+cosx)

(1-cosx)/IsinxI

Solve the system of linear equations. x+y−3z=−1 y−z=0 -x+2y=1

(2a-1,a,a)

Find the five key points (intercepts, maximum points, and minimum points) of the graph of y=2sin (x-π/4)

(5π/4, 0), (7π/4, -2), (9π/4, 0) (π/4, 0), (3π/4, 2)

Use a sum or difference formula to find the exact value of tan 255°

(9 + 6√3 + 3)/6