Trigonometry and Advanced Geometry

1. What is cos(180°)?

• B. 0.5
• C. -1 ✓
• D. 1

💡 The cosine of 180 degrees equals -1.

2. What is the value of sin²θ + cos²θ?

• B. 1 ✓
• C. 2
• D. undefined

💡 This is the Pythagorean identity in trigonometry, which always equals 1.

3. What is the formula for the circumference of a circle?

• A. πr²
• B. 2πr ✓
• C. πd²
• D. 2πr²

💡 The circumference of a circle is calculated as 2πr, where r is the radius.

4. How many equilateral triangles make up a regular hexagon when divided from its center?

• A. 4
• B. 5
• C. 6 ✓
• D. 7

💡 A regular hexagon can be divided into 6 equilateral triangles from its center.

5. In a right triangle with hypotenuse 10 and one leg 6, what is the other leg?

• A. 6
• B. 7
• C. 8 ✓
• D. 9

💡 Using the Pythagorean theorem: 10² - 6² = 100 - 36 = 64, and the square root of 64 is 8.

6. In a right triangle, what does the Pythagorean theorem state?

• A. a+b=c
• B. a²+b²=c² ✓
• C. a²-b²=c²
• D. a×b=c

💡 The Pythagorean theorem states that the sum of the squares of the two legs equals the square of the hypotenuse.

7. What is cos(0°)?

• B. 0.5
• C. 1 ✓
• D. undefined

💡 The cosine of 0 degrees equals 1.

8. What is sin(30°)?

• B. 0.5 ✓
• C. 0.87
• D. 1

💡 The sine of 30 degrees equals 0.5.

9. What is the volume of a sphere with radius 3 (use V=4/3πr³, π≈3.14)?

• A. 88
• B. 100
• C. 113 ✓
• D. 120

💡 Volume equals 4/3 x 3.14 x 27, which is approximately 113.

10. In a 30-60-90 triangle, what is the ratio of the sides?

• A. 1:1:2
• B. 1:√3:2 ✓
• C. 1:2:3
• D. 1:√2:2

💡 A 30-60-90 triangle has sides in the ratio 1:√3:2.

11. What is sin(90°)?

• B. 0.5
• C. 1 ✓
• D. undefined

💡 The sine of 90 degrees equals 1.

12. What is the sum of interior angles of a hexagon?

• A. 540
• B. 600
• C. 720 ✓
• D. 900

💡 Using the formula (n-2) x 180 for n=6 gives 720 degrees.

13. What is the value of sin(180°)?

• B. 0.5
• C. 1
• D. -1

💡 The sine of 180 degrees equals 0.

14. What is the value of π/2 radians in degrees?

• A. 45
• B. 90 ✓
• C. 180
• D. 270

💡 π/2 radians is equivalent to 90 degrees.

15. What is tan(45°)?

• B. 0.5
• C. 1 ✓
• D. undefined

💡 The tangent of 45 degrees equals 1, since sine and cosine are equal at this angle.

16. What is tan(0°)?

• B. 0.5
• C. 1
• D. undefined

💡 The tangent of 0 degrees equals 0.

17. What is the surface area formula for a sphere?

• A. 4πr² ✓
• B. πr²
• C. 2πr²
• D. 4/3πr³

💡 The surface area of a sphere is calculated using the formula 4πr².

18. What is cos(90°)?

• B. 0.5
• C. 1
• D. -1

💡 The cosine of 90 degrees equals 0.

19. What is the formula for the area of a trapezoid?

• A. 1/2(a+b)h ✓
• B. (a+b)h
• C. 1/2ah
• D. abh

💡 The area of a trapezoid is calculated as one-half the sum of the parallel sides, multiplied by the height.

20. What is the radius of a circle with circumference 62.8 (use π≈3.14)?

• A. 8
• B. 9
• C. 10 ✓
• D. 11

💡 Circumference equals 2πr, so r = 62.8 / (2 x 3.14) = 10.