📘 Precalculus Laws Of Sines Cosines (12 Focused Problems)
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #1
What is the value of $\sin(30°) + \cos(60°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{\sqrt{3}}{2}$ E) $\sqrt{2}$
Answer & Solution
Answer: C
Using exact values: $\sin(30°) = \frac{1}{2}$ and $\cos(60°) = \frac{1}{2}$, so $\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1$.
2.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #2
If $f(x) = 2^x$ and $g(x) = \log_2(x)$, what is $f(g(8))$?
A) $2$ B) $3$ C) $8$ D) $16$ E) $64$
Answer & Solution
Answer: C
First, $g(8) = \log_2(8) = 3$. Then $f(g(8)) = f(3) = 2^3 = 8$.
3.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #3
What is the period of $y = \sin(3x)$?
A) $\frac{\pi}{3}$ B) $\frac{2\pi}{3}$ C) $\pi$ D) $2\pi$ E) $3\pi$
Answer & Solution
Answer: B
For $y = \sin(bx)$, the period is $\frac{2\pi}{b}$. Here $b = 3$, so the period is $\frac{2\pi}{3}$.
4.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #4
If $z = 1 + i$, what is $|z|$?
A) $1$ B) $\sqrt{2}$ C) $2$ D) $\sqrt{3}$ E) $\sqrt{5}$
Answer & Solution
Answer: B
For $z = a + bi$, $|z| = \sqrt{a^2 + b^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.
5.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #5
What is the value of $\tan(45°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\sqrt{2}$ E) $\sqrt{3}$
Answer & Solution
Answer: C
Using the exact value, $\tan(45°) = 1$.
6.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #6
What is the value of $\sin(30°) + \cos(60°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{\sqrt{3}}{2}$ E) $\sqrt{2}$
Answer & Solution
Answer: C
Using exact values: $\sin(30°) = \frac{1}{2}$ and $\cos(60°) = \frac{1}{2}$, so $\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1$.
7.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #7
If $f(x) = 2^x$ and $g(x) = \log_2(x)$, what is $f(g(8))$?
A) $2$ B) $3$ C) $8$ D) $16$ E) $64$
Answer & Solution
Answer: C
First, $g(8) = \log_2(8) = 3$. Then $f(g(8)) = f(3) = 2^3 = 8$.
8.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #8
What is the period of $y = \sin(3x)$?
A) $\frac{\pi}{3}$ B) $\frac{2\pi}{3}$ C) $\pi$ D) $2\pi$ E) $3\pi$
Answer & Solution
Answer: B
For $y = \sin(bx)$, the period is $\frac{2\pi}{b}$. Here $b = 3$, so the period is $\frac{2\pi}{3}$.
9.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #9
If $z = 1 + i$, what is $|z|$?
A) $1$ B) $\sqrt{2}$ C) $2$ D) $\sqrt{3}$ E) $\sqrt{5}$
Answer & Solution
Answer: B
For $z = a + bi$, $|z| = \sqrt{a^2 + b^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.
10.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #10
What is the value of $\tan(45°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\sqrt{2}$ E) $\sqrt{3}$
Answer & Solution
Answer: C
Using the exact value, $\tan(45°) = 1$.
11.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #11
What is the value of $\sin(30°) + \cos(60°)$?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{\sqrt{3}}{2}$ E) $\sqrt{2}$
Answer & Solution
Answer: C
Using exact values: $\sin(30°) = \frac{1}{2}$ and $\cos(60°) = \frac{1}{2}$, so $\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1$.
12.
Tags: Precalculus · Difficulty (E/M/H) · source: AMC10 2020 #12
If $f(x) = 2^x$ and $g(x) = \log_2(x)$, what is $f(g(8))$?
A) $2$ B) $3$ C) $8$ D) $16$ E) $64$
Answer & Solution
Answer: C
First, $g(8) = \log_2(8) = 3$. Then $f(g(8)) = f(3) = 2^3 = 8$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | A | A | A | A | A | A | A | A | A | A | A |