Trigonometric Ratios Quiz: Math Trivia Test!

Explanation
To find sin(A), we need to use the trigonometric ratio of sine. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, since we only have the fraction 24/25 as the answer, we can assume that A is an acute angle in a right triangle. Therefore, the side opposite A would have a length of 24 and the hypotenuse would have a length of 25. Dividing the length of the opposite side by the length of the hypotenuse, we get 24/25 as the sine of angle A.

Explanation
The question is asking for the value of tan(B). The correct answer is 7/24. This means that the tangent of angle B is equal to 7/24.

Explanation
The correct answer is 12/13. This can be determined by using the definition of sine in a right triangle. Since sine is equal to the opposite side divided by the hypotenuse, we can see that in this case, the opposite side is 12 and the hypotenuse is 13. Therefore, sin(D) = 12/13.

Explanation
The correct answer is 5/12. This means that the tangent of angle E is equal to 5/12.

Explanation
The side opposite angle A is BC.

Explanation
The side opposite angle D is EF.

Explanation
Angle B is formed by the intersection of sides BC and AB. Therefore, the leg adjacent to angle B is BC.

Explanation
The tangent of 45 degrees is equal to 1. This can be derived from the fact that the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. In a 45-45-90 right triangle, the opposite and adjacent sides are equal in length, so the tangent is equal to 1.

Explanation
The correct answer is 1/sqrt(2) because the cosine of 45 degrees is equal to the adjacent side divided by the hypotenuse in a right triangle. In a triangle with a 45-degree angle, the adjacent side and the hypotenuse are equal in length. By using the Pythagorean theorem, we can find that the length of both sides is sqrt(2). Therefore, the cosine of 45 degrees is 1/sqrt(2).

Explanation
The statement is true because the sine of 45 degrees is equal to the cosine of 45 degrees. This is because the sine function and the cosine function are equal for complementary angles. In a right triangle, the sine of an angle is equal to the cosine of its complement, and vice versa. Since 45 degrees and its complement 45 degrees are equal, their sine and cosine values are also equal.

Explanation
The value of tan(45) can be found without a calculator because it is one of the special angles in trigonometry. The exact value of tan(45) is 1. This is because in a right triangle with a 45-degree angle, the opposite side and adjacent side are equal, making the tangent ratio equal to 1.

Explanation
The cosine of 60 degrees is equal to 1/2. This can be determined using the unit circle or by using the cosine function. In the unit circle, at 60 degrees, the x-coordinate of the point on the unit circle is 1/2. Therefore, the cosine of 60 degrees is 1/2.

Explanation
The value of tan(30) can be found using the special right triangle with angles 30-60-90. In this triangle, the side opposite the 30-degree angle is 1, the side opposite the 60-degree angle is sqrt(3), and the hypotenuse is 2. The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, tan(30) = 1/sqrt(3), as the side opposite the 30-degree angle is 1 and the side adjacent to the angle is sqrt(3).

Explanation
The sine of 60 degrees is equal to the square root of 3 divided by 2.

Explanation
The statement sin(30) = cos(30) is false. In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse, while the cosine of an angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. In a 30-60-90 triangle, the sine of 30 degrees is 1/2, while the cosine of 30 degrees is √3/2. Therefore, sin(30) is not equal to cos(30).

Explanation
The given statement is true because the sine of 30 degrees is equal to the cosine of 60 degrees. In a right-angled triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse, while the cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse. In a 30-60-90 triangle, the side opposite the 30 degree angle is equal to half the length of the hypotenuse, and the side adjacent to the 60 degree angle is equal to half the length of the hypotenuse. Therefore, sin(30) = cos(60).

Explanation
The statement is false because you do not need a calculator to find the value of cos(60). The value of cos(60) is a well-known value in trigonometry, which is equal to 0.5. Therefore, you can easily determine the value without the need for a calculator.