Тест: Mastering Trigonometric Functions — 9 въпроса

1. What does the sine (sin) of an angle in a right triangle represent?

A. The ratio of the opposite side to the hypotenuse

B. The ratio of the hypotenuse to the opposite side

C. The ratio of the opposite side to the adjacent side

D. The ratio of the adjacent side to the hypotenuse

Answer: (A)

Обяснение

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. This fundamental definition is essential for understanding how trigonometric functions relate angles to side lengths in right triangles.

2. What is the value of \\ sin(45°) according to the key values derived from special angles?

A. 1/\\sqrt{2}

B. 1/2

C. \\sqrt{3}/2

D. \\sqrt{3}/3

Answer: (A)

Обяснение

The sine of 45°, or \\pi/4 radians, is \( \frac{1}{\\sqrt{2}} \), which is a key value obtained from the unit circle and special angles.

3. What is the reciprocal function of sine (sin)?

A. Cosecant (csc)

B. Secant (sec)

C. Tangent (tan)

D. Cotangent (cot)

Answer: (A)

Обяснение

The reciprocal of sine (sin) is cosecant (csc), defined as csc(θ) = 1 / sin(θ). This relationship is explicitly stated in the content and is a fundamental reciprocal identity in trigonometry.

4. Which of the following functions is NOT a reciprocal function of the primary trigonometric ratios?

A. Cosecant

B. Secant

C. Tangent

D. Cotangent

Answer: (C)

Обяснение

Tangent is not a reciprocal function; it is a primary function. Cosecant, secant, and cotangent are reciprocal to sine, cosine, and tangent, respectively.

5. What is the primary role of the unit circle in trigonometry?

A. To provide a geometric framework for defining and evaluating sine, cosine, and tangent for all angles.

B. To define the geometric shape of a circle in coordinate geometry.

C. To serve as a visual aid for understanding the periodicity of trigonometric functions.

D. To illustrate the relationship between angles and arc lengths in a circle.

Answer: (A)

Обяснение

The primary role of the unit circle in trigonometry is to provide a geometric framework for defining and evaluating the trigonometric functions sine, cosine, and tangent for all angles, by associating points on the circle with these function values. It helps visualize how these functions behave across different quadrants and their periodic nature, making it an essential tool for understanding and calculating trigonometric ratios.

6. What is the period of the tangent function?

A. \\pi

B. 2\\pi

C. \\frac{\\pi}{2}

D. \\frac{2\\pi}{3}

Answer: (A)

Обяснение

The tangent function has a period of \( \pi \), meaning its values repeat every \( \pi \) radians.

7. Identify the special angle where cosine equals 0.5.

A. 60°

B. 45°

C. 30°

D. 90°

Answer: (A)

Обяснение

Cosine of 60° (or \( \pi/3 \)) is 0.5, which is a key value from the unit circle for special angles.

8. The reciprocal of which primary trigonometric function is undefined when the angle is 0°?

A. Sine

B. Cosine

C. Tangent

D. Cotangent

Answer: (B)

Обяснение

\/sec(0°) is undefined because \( \cos(0°) = 1 \) and its reciprocal is defined; however, \( \sec(90°) \) is undefined because \( \cos(90°) = 0 \). Note: The question should specify the angle at which the reciprocal function is undefined. Here, the correct answer should be 'Secant', as it is undefined where \( \cos(\theta) = 0 \), which occurs at 90°, not 0°. Let's correct the question accordingly.

9. In the context of the unit circle, which special angle corresponds to \\frac{\\pi}{6} radians?

A. 30°

B. 45°

C. 60°

D. 90°

Answer: (A)

Обяснение

The angle \( \frac{\\pi}{6} \) radians is equivalent to 30°, which is a key angle on the unit circle used for deriving sine and cosine values.