📘 Precalculus Warmups
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Basic Trigonometry · Easy · source: Original (AMC-style)
What is $\sin(30°)$?
A) $\frac{1}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $\frac{\sqrt{3}}{2}$ D) $1$ E) $\frac{1}{\sqrt{2}}$
Answer & Solution
Answer: A
$\sin(30°) = \frac{1}{2}$.
2.
Tags: Cosine Values · Easy · source: Original (AMC-style)
What is $\cos(60°)$?
A) $\frac{1}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $\frac{\sqrt{3}}{2}$ D) $1$ E) $\frac{1}{\sqrt{2}}$
Answer & Solution
Answer: A
$\cos(60°) = \frac{1}{2}$.
3.
Tags: Tangent Values · Easy · source: Original (AMC-style)
What is $\tan(45°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\sqrt{2}$ E) $\sqrt{3}$
Answer & Solution
Answer: C
$\tan(45°) = 1$.
4.
Tags: Pythagorean Identity · Easy · source: Original (AMC-style)
What is $\sin^2(30°) + \cos^2(30°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{3}{2}$ E) $2$
Answer & Solution
Answer: C
By the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$ for any angle $\theta$.
5.
Tags: Unit Circle · Easy · source: Original (AMC-style)
What is $\sin(90°)$?
A) $0$ B) $\frac{1}{2}$ C) $\frac{\sqrt{2}}{2}$ D) $1$ E) $\frac{\sqrt{3}}{2}$
Answer & Solution
Answer: D
$\sin(90°) = 1$.
6.
Tags: Reference Angles · Easy · source: Original (AMC-style)
What is $\cos(120°)$?
A) $-\frac{1}{2}$ B) $\frac{1}{2}$ C) $-\frac{\sqrt{3}}{2}$ D) $\frac{\sqrt{3}}{2}$ E) $-1$
Answer & Solution
Answer: A
$\cos(120°) = \cos(180° - 60°) = -\cos(60°) = -\frac{1}{2}$.
7.
Tags: Trig Identities · Easy · source: Original (AMC-style)
What is $\sec(0°)$?
A) $0$ B) $1$ C) $\frac{1}{2}$ D) $2$ E) Undefined
Answer & Solution
Answer: B
$\sec(0°) = \frac{1}{\cos(0°)} = \frac{1}{1} = 1$.
8.
Tags: Inverse Trig · Easy · source: Original (AMC-style)
What is $\arcsin(\frac{1}{2})$?
A) $15°$ B) $30°$ C) $45°$ D) $60°$ E) $90°$
Answer & Solution
Answer: B
$\arcsin(\frac{1}{2}) = 30°$ because $\sin(30°) = \frac{1}{2}$.
9.
Tags: Complex Numbers · Easy · source: Original (AMC-style)
What is $i^2$?
A) $1$ B) $-1$ C) $i$ D) $-i$ E) $0$
Answer & Solution
Answer: B
By definition, $i^2 = -1$.
10.
Tags: Complex Addition · Easy · source: Original (AMC-style)
What is $(2+3i) + (1-2i)$?
A) $3+i$ B) $3-i$ C) $1+5i$ D) $1+i$ E) $3+5i$
Answer & Solution
Answer: A
$(2+3i) + (1-2i) = (2+1) + (3-2)i = 3+i$.
11.
Tags: Complex Multiplication · Easy · source: Original (AMC-style)
What is $(1+i)(1-i)$?
A) $0$ B) $1$ C) $2$ D) $2i$ E) $1+i$
Answer & Solution
Answer: C
$(1+i)(1-i) = 1 - i^2 = 1 - (-1) = 2$.
12.
Tags: Exponential Functions · Easy · source: Original (AMC-style)
What is $2^0$?
A) $0$ B) $1$ C) $2$ D) $4$ E) Undefined
Answer & Solution
Answer: B
Any non-zero number to the power of 0 is 1.
13.
Tags: Logarithmic Functions · Easy · source: Original (AMC-style)
What is $\log_2(8)$?
A) $2$ B) $3$ C) $4$ D) $8$ E) $16$
Answer & Solution
Answer: B
Since $2^3 = 8$, we have $\log_2(8) = 3$.
14.
Tags: Sequences · Easy · source: Original (AMC-style)
What is the 5th term of the arithmetic sequence 2, 5, 8, 11, …?
A) $14$ B) $15$ C) $16$ D) $17$ E) $18$
Answer & Solution
Answer: A
The common difference is 3, so the 5th term is $11 + 3 = 14$.
15.
Tags: Series · Easy · source: Original (AMC-style)
What is the sum of the first 4 terms of 1, 2, 4, 8, …?
A) $10$ B) $12$ C) $15$ D) $16$ E) $20$
Answer & Solution
Answer: C
The sum is $1 + 2 + 4 + 8 = 15$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | A | C | C | D | A | B | B | B | A | C | B | B | A | C |