Please answer the following for advanced functions I am very confused

Question

Question 1 [KU]

The point P(1/2, -√3/2) lies on the unit circle.

a. Sketch the unit circle showing the point P and the standard position angle θ drawn to the terminal arm containing point P.

b. Determine the exact values of the six trigonometric ratios for θ.

c. Recall that, if the value of a particular trigonometric ratio for some angle θ is known, the corresponding inverse trigonometric function can be used to determine the measure of θ.

Using one of the trigonometric ratios from part (b), calculate the measure of angle θ in both degrees and radians.

d. Using the sketch from part (a), identify the coordinates of the point on the unit circle that is also on the terminal arm of standard position angle θ = -θ.

Question 2 [TI]

a. If csc(θ) = 2 and tan(θ) < 0, determine the exact value of cos(θ).

b. Without using a calculator, determine the value of θ, where θ is an acute angle.

tan(π/2 − θ) = cot(θ)

Question 3 [APP]

Ashley is riding a Ferris wheel that has a diameter of 40 metres. The wheel revolves at a rate of 1.5 revolutions per minute. Ashley’s height above the ground, h, after t minutes can be given by the equation:

h = 21 − 20 cos(3πt)

a. What is Ashley’s maximum height above the ground while she is riding the Ferris wheel?

b. If Ashley gets on the ride at its lowest point, how high above the ground will she be to start the ride?

c. How many times does Ashley go around the Ferris wheel in four minutes?

d. Draw a sketch of the rider’s height above the ground at any time during the first four minutes.

e. How long after she starts riding will her height be 31 metres above the ground?

f. In the four minutes that she spends riding the Ferris wheel, what is the total amount of time that Ashley’s height is above 31 metres?

Question 4 [COMM]

Prove the following identities:

a. cos(θ) + tan(θ) · sin(θ) = sec(θ)

b. cot(x) + sin(x) / (1 + cos(x)) = csc(x)

Answer 1

1) (1/2, -sqr3/2) lies on a unit circle x^2 +y^2 = 1

1/4 + 3/4 = 1

angle = arctan(sqr3/2/1/2) = 60 degrees = pi/3 radians

2) cscT=2 =1/cosT, cosT =1/2, T =arccos.5 = 60 degrees = pi/3 radians

3) use a sinusoidal function to solve

4) see other posted answer

Answer 2

Here is an answer to question 4, part a)

To prove a trig identity pick a side to work on and make it look like the other side.

cos x + tan x (sin x) ≡ sec x (let's work on the left side)

cos x + sinx/cos x (sin x) (replace tan x with sinx / cos x)

cos x + sin²x / cos x (since there are two terms on the left, one on the right, let's get a common denominator and add)

[cos²x + sin²x]/cos x (now replace the numerator with 1)

1 / cos x (used the Pythagorean Identity sin² + cos² =1 -- always)

sec x ≡ sec x (used reciprocal identity for cos)

Q.E.D

(or some teacher's suggest a smiley face, something to show that the proof is complete)