Math Final

Convert (2,𝜋3) to rectangular coordinates.

(1,√3)

Simplify the given expression. cos(−𝑥) sin𝑥 cot𝑥+sin^2𝑥

cos^2𝑥+sin^2𝑥

Which set of polar coordinates most closely matches the point plotted on the graph?

(2√2, 𝜋/4)

Find the exact value. tan(3𝜋/8)

1+√2

Rewrite the expression sin^4𝑥 with no powers greater than 1.

1/8(3+cos(4𝑥)−4cos(2𝑥))

Find the area of the triangle. Round each answer to the nearest tenth.

15.2

A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15 degrees to the right of his original course and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

1712 miles

Given 𝑧1=8cis(36∘) and 𝑧2=2cis(15∘), evaluate this expression. √z1

2√2 𝑐𝑖𝑠(18∘), 2√2 𝑐𝑖𝑠(198∘

Find sin(𝜃2), cos(𝜃2), and tan(𝜃2) given c os𝜃=7/25 and 𝜃 is in quadrant IV.

3/5,−4/5,−3/4

Given 𝑧1=8cis(36∘) and 𝑧2=2cis(15∘), evaluate this expression. 𝑧1/𝑧2

4cis (21∘)

The displacement ℎ(𝑡) in centimeters of a mass suspended by a spring is modeled by the function ℎ(𝑡)=1/4sin(120𝜋𝑡),where 𝑡 is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Amplitude: 1/4 period 1/60, frequency: 60 Hz

Two frequencies of sound are played on an instrument governed by the equation 𝑛(𝑡)=8cos(20𝜋𝑡)cos(1000𝜋𝑡).What are the period and frequency of the "fast" and "slow" oscillations? What is the amplitude? Hint: slowperiod=2𝜋20𝜋=110 fastperiod=2𝜋1000𝜋=1500

Amplitude: 8, fast period: 1500, fast frequency: 500 Hz, slow period: 110, slow frequency: 10 Hz

A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal, the ball is released 3.5 feet above the ground. Additional info: 𝑣0=95ft/s 𝜃=52∘ ℎ=3.5 𝑔=32ft/s2 Where is the ball after 2 seconds?

The ball is 89.2 feet high and 117 feet from where it was launched.

Prove the identity. tan^3𝑥−tan𝑥 sec^2𝑥 = tan(−𝑥)

tan(−𝑥)=tan(−𝑥)

Convert this graph and Cartesian equation to rectangular form. 𝑟=−3 csc𝜃

y=-3

Find sin(2𝜃), cos(2𝜃), and tan(2𝜃) given cot𝜃=−3/4 and 𝜃 is on the interval [𝜋/2,𝜋].

−24/25, −7/25, 24/7

5cos(2𝜋/3) + 5𝑖 sin(2𝜋/3)

−5/2 + 𝑖5√3/2

Which Cartesian equation is best represented by the graph?

𝑟= 3 − 5𝑠𝑖𝑛𝜃

Which Cartesian equation is best represented by the graph?

𝑟= −3 + 3𝑐𝑜𝑠𝜃

Convert the polar equation to a Cartesian equation. 𝑥^2+𝑦^2=5𝑦

𝑟=5 𝑠𝑖𝑛𝜃

Simplify the given expression. sin(−𝑥)cos(−2𝑥)−sin(−𝑥)cos(−2𝑥)

0

Find all exact solutions to the equation on [0,2𝜋). cos^2𝑥=cos𝑥

0, 𝜋/2, 3𝜋/2

Find all exact solutions to the equation on [0,2𝜋). 2sin^2𝑥−sin𝑥=0

0, 𝜋/6, 5𝜋/6, 𝜋

Find all exact solutions to the equation on [0,2𝜋). cos^2𝑥−sin^2𝑥−1=0

0,𝜋

Given 𝑧1= 8cis(36∘) and 𝑧2= 2cis(15∘), evaluate this expression. (𝑧2)^3

8cis (45∘)

Find the absolute value of the complex number. 5−9𝑖

√106

Write the complex number in polar form. 4+𝑖

√17 cis(14∘)1

Find the exact value. 2sin(𝜋/4)sin(𝜋/6)

√2/2

Find the exact value. cos(7𝜋/12)

√2−√6/4

Plot the points and find a function of the form 𝑦=𝐴 cos(𝐵𝑥+𝐶)+𝐷 that fits the given data.

𝑦=2 cos(𝜋𝑥+𝜋)

Eliminate the parameter 𝑡t to rewrite the following parametric equations as a Cartesian equation: {𝑥(𝑡)=𝑡+1. {𝑦(𝑡)=2𝑡^2

𝑦=2(𝑥−1)^2

Assume 𝛼α is opposite side 𝑎, 𝛽 is opposite side 𝑏, and 𝛾 is opposite side 𝑐. Solve the triangle, if possible. Round each answer to the nearest tenth.𝛽=68∘,𝑏=21,𝑐=16

𝛼=67.1∘, 𝛾=44.9∘, 𝑎=20.9

Find all solutions of tan(𝑥)−√3=0.

𝜋/3+𝑘𝜋