Q: Express $$\sin^4$ \theta in $$terms of cosine functions without even powers.

$$\frac{3}{8} - \frac{1}{2} \cos(2\theta) + \frac{1}{8} \cos(4\theta)$$

Q: Express $$a \cos x + b \sin x in $$the form $$R \cos(x - \theta). If$$ $$a = 1$$ and $$b = \sqrt{3}$$, what are $$R$$ and $$\theta$$?

$$R = 2, \theta = \frac{\pi}{3}$$

Question 1

Which of the following statements correctly describes the relationship between rational and irrational numbers?

Accepted Answer

The set of real numbers consists of both rational and irrational numbers.

Question 2

Given the function $$f(x) = \frac{1}{\sqrt{x-1}}$$, what is its natural domain?

Accepted Answer

$$(1, \infty)$$

Question 3

Solve the inequality $$|2x + 4| \u003c 1$$ and express the solution in interval notation.

Accepted Answer

$$(-5/2, -3/2)$$

Question 4

Which of the following functions is odd?

Accepted Answer

$$f(x) = x - x^3$

Question 5

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4. $$What is the domain of the composite function $$f \circ g$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 6

A car is moving at a rate of $$80 km/h. $$The diameter of its wheels is $$60 cm. $$How many revolutions per minute do the wheels make?

Accepted Answer

$$425.5$$

Question 7

Find the area of a sector with radius $$3$$ and subtending an angle of $$\frac{3\pi}{5}$$ radians.

Accepted Answer

$$\frac{9\pi}{5}$$

Question 8

Which of the following is the correct period for the function $$	an(x)$$?

Accepted Answer

$$\pi$$

Question 9

Express $$\sin^4$ 	heta in $$terms of cosine functions without even powers.

Accepted Answer

$$\frac{3}{8} - \frac{1}{2} \cos(2	heta) + \frac{1}{8} \cos(4	heta)$$

Question 10

Find the polar coordinates of the point $$(x, y) = (-3, \sqrt{3}).$$

Accepted Answer

$$(2\sqrt{3}, \frac{5\pi}{6})$$

Question 11

Convert the polar equation $$r = 2 \sin 	heta to $$Cartesian form.

Accepted Answer

$$x^2 + (y$$ - $$1)^2 = 1$

Question 12

Express $$a \cos x + b \sin x in $$the form $$R \cos(x - 	heta). If$$ $$a = 1$$ and $$b = \sqrt{3}$$, what are $$R$$ and $$	heta$$?

Accepted Answer

$$R = 2, 	heta = \frac{\pi}{3}$$

Question 13

Use mathematical induction to prove that $$1 + 3 + 5 + \cdots + (2n - 1) = n^2$ for all n$$ \in \mathbb{N}$$. What is the value of the sum when n = 5$?

Accepted Answer

$$25$$

Question 14

Evaluate $$\sum_{r=1}^{n} r^2$ using the formula for the sum of squares.

Accepted Answer

$$\frac{n(n+1)(2n+1)}{6}$$

Question 15

Expand $$(a + b)^4$ using the binomial theorem.

Accepted Answer

$$a^4$$ + $$4a^3b$$ + $$6a^2b^2$ + 4ab^3$ + b^4$

Question 16

Which of the following conic sections is represented by the equation $$x^2$$ + $$y^2 = 4$?

Accepted Answer

Circle

Question 17

Given the quadratic form $$Ax^2 + $$Bxy + $$Cy^2 + Dx + Ey + F = 0$, which transformation is required to eliminate the $$Bxy$$ term?

Accepted Answer

Rotation of axes

Question 18

If $$f(x) = x^3$ and g(x$$) = $$x^2$$, which function is one-to-one?

Accepted Answer

$$f(x)$$

Comprehensive Study Notes for College Algebra: Numbers, Functions, Trigonometry, Induction, Sigma Notation, and Conic Sections

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