Trigonometric Formulas, Ratios, and Practice Questions
Basic Trigonometric Ratios
For a right-angled triangle with an angle $\theta$:
- $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
- $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
- $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
- $\csc(\theta) = \frac{1}{\sin(\theta)}$
- $\sec(\theta) = \frac{1}{\cos(\theta)}$
- $\cot(\theta) = \frac{1}{\tan(\theta)}$
Fundamental Identities
Pythagorean Identities:
- $\sin^2(\theta) + \cos^2(\theta) = 1$
- $1 + \tan^2(\theta) = \sec^2(\theta)$
- $1 + \cot^2(\theta) = \csc^2(\theta)$
10 Multiple Choice Questions
1. What is the exact value of $\sin(30^\circ)$?
- A) $0$
- B) $\frac{1}{2}$ ✓ Correct
- C) $\frac{\sqrt{2}}{2}$
- D) $1$
Solution: In a standard $30^\circ-60^\circ-90^\circ$ right triangle, the ratio of the side opposite the $30^\circ$ angle to the hypotenuse is exactly $\frac{1}{2}$. Therefore, $\sin(30^\circ) = \frac{1}{2}$.
2. Which of the following expressions is equivalent to $\tan(\theta)$?
- A) $\frac{\cos(\theta)}{\sin(\theta)}$
- B) $\frac{\sin(\theta)}{\cos(\theta)}$ ✓ Correct
- C) $\frac{1}{\sin(\theta)}$
- D) $\sin(\theta)\cos(\theta)$
Solution: By definition, the tangent of an angle is the ratio of its sine to its cosine. So, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.
3. Simplify the trigonometric expression: $\sin^2(x) + \cos^2(x)$.
- A) $0$
- B) $\tan^2(x)$
- C) $1$ ✓ Correct
- D) $2\sin(x)$
Solution: This represents the fundamental Pythagorean identity in trigonometry, which dictates that for any angle $x$, $\sin^2(x) + \cos^2(x) = 1$.
4. What is the equivalent value of $\cos(90^\circ - \theta)$?
- A) $\sin(\theta)$ ✓ Correct
- B) $-\sin(\theta)$
- C) $\cos(\theta)$
- D) $-\cos(\theta)$
Solution: This utilizes the co-function identity. The cosine of complementary angles is equal to the sine of the angle itself, meaning $\cos(90^\circ - \theta) = \sin(\theta)$.
5. What is the period of the standard sine function, $y = \sin(x)$?
- A) $\pi$
- B) $2\pi$ ✓ Correct
- C) $\frac{\pi}{2}$
- D) $4\pi$
Solution: The sine wave function completes one full, repetitive cycle over the interval of $2\pi$ radians. Therefore, its period is $2\pi$.
6. Evaluate the exact value of $\tan(45^\circ)$.
- A) $0$
- B) $\frac{1}{2}$
- C) $\sqrt{2}$
- D) $1$ ✓ Correct
Solution: In an isosceles right triangle ($45^\circ-45^\circ-90^\circ$), the opposite and adjacent legs are equal in length. Because tangent is opposite over adjacent, $\tan(45^\circ) = \frac{x}{x} = 1$.
7. Identify the correct double-angle formula for $\sin(2\theta)$.
- A) $\cos^2(\theta) - \sin^2(\theta)$
- B) $2\sin(\theta)\cos(\theta)$ ✓ Correct
- C) $1 - 2\sin^2(\theta)$
- D) $2\cos(\theta)$
Solution: The double angle identity for sine is derived from the angle addition formula: $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$. By setting $A = B = \theta$, it simplifies to $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.
8. Given that $\sec(\theta) = 2$, calculate the value of $\cos(\theta)$.
- A) $2$
- B) $\frac{1}{2}$ ✓ Correct
- C) $\frac{\sqrt{3}}{2}$
- D) $-2$
Solution: Secant is defined as the reciprocal of cosine ($\sec(\theta) = \frac{1}{\cos(\theta)}$). If $\sec(\theta) = 2$, then taking the reciprocal of both sides gives $\cos(\theta) = \frac{1}{2}$.
9. Convert the radian measure $\frac{\pi}{4}$ to degrees.
- A) $30^\circ$
- B) $45^\circ$ ✓ Correct
- C) $60^\circ$
- D) $90^\circ$
Solution: To convert a measure from radians to degrees, multiply the value by the conversion factor $\frac{180^\circ}{\pi}$. Therefore, $\frac{\pi}{4} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{4} = 45^\circ$.
10. Based on the Law of Sines for a triangle with sides $a, b, c$ and opposite angles $A, B, C$ respectively, which of the following equations is true?
- A) $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ ✓ Correct
- B) $a^2 = b^2 + c^2 - 2bc\cos(A)$
- C) $\sin(A) + \sin(B) = \sin(C)$
- D) $\frac{\sin(a)}{A} = \frac{\sin(b)}{B}$
Solution: The Law of Sines dictates that the proportion between the length of a side of any triangle to the sine of the angle directly opposite to it remains constant for all three sides and angles.