📘 Precalculus Trigonometry Basics
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Basic Trig · 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°) = 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°) = 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: Special Angles · 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.
5.
Tags: Trig Identities · Easy · source: Original (AMC-style)
What is sin²(30°) + cos²(30°)?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{3}{2}$ E) $2$
Answer & Solution
Answer: C
By the Pythagorean identity: sin²(θ) + cos²(θ) = 1 for any angle θ.
6.
Tags: Trig Ratios · Easy · source: Original (AMC-style)
In a right triangle with angle 30°, if the opposite side is 1, what is the hypotenuse?
A) $1$ B) $2$ C) $\sqrt{2}$ D) $\sqrt{3}$ E) $\frac{1}{2}$
Answer & Solution
Answer: B
sin(30°) = opposite/hypotenuse = 1/hypotenuse = 1/2, so hypotenuse = 2.
7.
Tags: Trig Functions · Medium · source: Original (AMC-style)
What is the period of sin(x)?
A) $\pi$ B) $2\pi$ C) $4\pi$ D) $\frac{\pi}{2}$ E) $\frac{\pi}{4}$
Answer & Solution
Answer: B
The period of sin(x) is 2Ï€.
8.
Tags: Trig Graphs · Medium · source: Original (AMC-style)
What is the amplitude of 3sin(x)?
A) $1$ B) $2$ C) $3$ D) $6$ E) $9$
Answer & Solution
Answer: C
The amplitude of 3sin(x) is 3.
9.
Tags: Trig Equations · Medium · source: Original (AMC-style)
What is the general solution to sin(x) = 0?
A) $x = 0$ B) $x = \pi$ C) $x = n\pi$ for integer n D) $x = \frac{\pi}{2} + n\pi$ for integer n E) $x = 2n\pi$ for integer n
Answer & Solution
Answer: C
sin(x) = 0 when x = nπ for any integer n.
10.
Tags: Trig Identities · Medium · source: Original (AMC-style)
What is 1 + tan²(45°)?
A) $1$ B) $2$ C) $\sqrt{2}$ D) $\frac{1}{2}$ E) $\frac{3}{2}$
Answer & Solution
Answer: B
1 + tan²(45°) = 1 + 1² = 2.
11.
Tags: Advanced Trig · Hard · source: Original (AMC-style)
What is sin(15°)?
A) $\frac{\sqrt{6} - \sqrt{2}}{4}$ B) $\frac{\sqrt{6} + \sqrt{2}}{4}$ C) $\frac{\sqrt{3} - 1}{2}$ D) $\frac{\sqrt{3} + 1}{2}$ E) $\frac{1}{2}$
Answer & Solution
Answer: A
Using the angle difference formula: sin(15°) = sin(45° - 30°) = (√6 - √2)/4.
12.
Tags: Complex Trig · Hard · source: Original (AMC-style)
What is the value of sin(75°)cos(15°)?
A) $\frac{1}{4}$ B) $\frac{1}{2}$ C) $\frac{\sqrt{2}}{4}$ D) $\frac{\sqrt{3}}{4}$ E) $\frac{1}{8}$
Answer & Solution
Answer: A
Using the product-to-sum formula: sin(75°)cos(15°) = (1/2)[sin(90°) + sin(60°)] = (1/2)[1 + √3/2] = 1/4.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | A | C | D | C | B | B | C | C | B | A | A |