Mathematics Quiz

Which of the following is true for the expression \( \sin(\pi - a) \)?

Answer explanation

Using the sine subtraction formula, \( \sin(\pi - a) = \sin(a) \). Therefore, if \( a = \frac{7\pi}{6} \), then \( \sin(a) = -\frac{1}{2} \). Thus, the correct answer is \( \sin(\pi - a) = -\frac{1}{2} \).

• I

• II

• III

• IV

Answer explanation

To find the quadrant of point M, we reduce the angle 510° by subtracting 360°, resulting in 150°. Since 150° is in the range of 90° to 180°, point M lies in the II quadrant.

Answer explanation

Given \( m = \tan \alpha + \cot \alpha \), we can square both sides: \( m^2 = \tan^2 \alpha + 2 + \cot^2 \alpha \). Thus, \( \tan^2 \alpha + \cot^2 \alpha = m^2 - 2 \), leading to the answer \( m^2 - 2 \).

Answer explanation

Since \(\alpha \) is in the range \(\frac{4}{5} \pi < \alpha < \pi\), it is in the second quadrant where cosine is negative. The reference angle is \(\alpha - \pi\), leading to \(\cos \alpha = -\frac{3}{5}\).

In triangle \( ABC \), which of the following identities is always true?

• \( \sin(A + B) = \sin C \)

Answer explanation

In triangle ABC, the angles A, B, and C satisfy A + B + C = 180°. Thus, A + B = 180° - C. Using the sine and cosine identities, we find that \( \sin(A + B) = \sin(180° - C) = \sin C \), making the second choice correct.

Answer explanation

The angle given is \( \frac{-\pi}{5} + k2\pi \). For \( k=0 \), the angle is \( \frac{-\pi}{5} \), which is the correct measure. Other choices represent different angles but do not match the specified angle.

• 4 points

• 5 points

• 6 points

• 7 points

Answer explanation

The angle µ = \frac{-\pi}{5} + \frac{k\pi}{3} represents points on the unit circle for integer values of k. The period of the angle is 2\pi, and the least common multiple of the denominators (5 and 3) is 15. Thus, there are 6 distinct points.

What is the area of the polygon represented by the angle $ \alpha $ on the unit circle?

• 1

• 2

• 3

• 4

Answer explanation

The area of the polygon formed by angle \( \alpha \) on the unit circle is calculated as \( \frac{1}{2} r^2 \theta \). For a unit circle (\( r=1 \)), the area simplifies to \( \frac{1}{2} \times 2 = 1 \). However, the polygon formed by two angles gives an area of 2.

Answer explanation

To find the angle, substitute k = -1 into \( \frac{30\pi}{7} + k2\pi \), giving \( \frac{30\pi}{7} - 2\pi = \frac{30\pi}{7} - \frac{14\pi}{7} = \frac{16\pi}{7} \). Since angles are periodic, reduce it to \( \frac{2\pi}{7} \).

• 2 points

• 3 points

• 4 points

• 5 points

Answer explanation

The angle \( \frac{30\pi}{7} + \frac{k2\pi}{3} \) can be simplified to find equivalent angles on the unit circle. The period of the angle is \( 2\pi \), and there are 3 distinct angles within one full rotation, leading to 3 points.